The Comprehensive, Exhaustive and Exhausting Rundown of Number None's Magic System Deep Lore, Sponsored By Brandon Sanderson
The following is going to be a mix of actual canon, my own extrapolations, and some changes, but in essence...
The differences between the various types of supernatural beings largely come down to the question of "How do I channel the Heart?"
Every soul has an innate power that is shaped by their core self and sense of identity. That's the Heart. This is unrelated to spiritual power; everyone has one, even the weakest Rukongai brat or cubicle worker. However, it is generally impossible to access the Heart. It is part of one's fundamental being, and cannot be manifested through training or willpower. People like Orihime or Tatsuki are humans who through abnormal circumstances, such as long-term exposure to tremendous amounts of reiatsu, living on Spirit Grounds all their life, and proximity to the Hogyokou for months, eventually accessed their Heart and gained the ability to manifest it through a physical or abstract medium - in Orihime's case, her hairpin, and in Tatsuki's case, her martial arts.
But they're freak accidents. It is impossible to consistently reproduce such results and give a soul natural access to the Heart. So other beings have their own methods.
Hollows and How The Only Natural Way To Do It Will Also Make You A Monster Because This Universe Sucks
A Hollow literally rips their Heart out of their soul, externalizing it, and then shapes it into a body. This grants them immediate access to their inner identity-based power, as well as manifesting it in a physical fashion that grants additional rewards in terms of increased strength and toughness, at the low low cost of having, you know, turned their emotional core into a suit and thus no longer having it inside their own soul, which makes them crazy cannibalistic monsters.
The body of a Hollow is essentially a giant suit. The person is there, inside it (not necessarily in their physical human shape, though that sometimes happens), literally wearing their Heart as an armor against the world.
Shinigami and Cheating Your Way Into Superpower
Shinigami use a gentler form of a similar process. Their swords are effectively blank souls, and as they wield them, they create a bond, but the sword does not inherently have a personality or self. What happens is that the soul's Heart "migrates" from their own self into the blank template of the Hollow-like sword, reshaping it in accordance with the wielder's core identity. This is what is meant by "The blade is me;" a Zanpakutou is a Shinigami's innermost self, manifested through an external channel. Shinigami often talk about not being able to hear one's sword because one has not yet forged a strong enough bond, but that's not the whole picture. The Asauchi is a soul, but it is mindless. When the bond is forged between the Shinigami's soul and that blank Hollow, the Heart crosses over into the sword, and the mindless ghost becomes a sapient being.
Shinigami are the only beings to have powers with a personality because they are the only ones who actually use a secondary soul as a vector to manifest their Heart. Even Orihime's fairies aren't really separate souls; they are self-willed being but they're part of Orihime's soul, not metaphysically independant (though that is a distinction without relevant consequences).
When Rukia transfers her powers to Ichigo, she effectively gifts him with part of her own Heart along with the blank soul it has imprinted on, but since Ichigo is not Rukia and has no bond with this sword, it never fully crystallizes into a second Sode no Shirayuki. His Zanpakutou is technically Sode no, but it fails to form itself into a sword-spirit because of their lack of connection, and so "she" is not actually self-aware. This is a good thing because when Ichigo finally bonds with his sword during Urahara's training, that initial Heart is stamped over with Ichigo's own, creating Zangetsu, and that would have been murder.
Arrancars and The True Loneliness Of The Soul
The above is why Arrancars don't have sword-spirits. The process of Arrancarization consists in taking the mask (and by extension, the whole armor-body) of a Hollow and breaking it to reveal the person inside. That mask and that body are the Hollow's Heart, externalized; if the process goes correctly, they reshape themselves into another externalized form, another channel. In theory this could be any kind of object or garment, it's only swords because this is a ghost samurai story. Thus, the Arrancar is born with ready access to their Heart, and their release causes their sword to return to (something resembling) its original personality-related, armor-suit-body state. But because they actually have a proper bond with their Heart, the Arrancar is no longer missing their emotional core and is no longer driven to fill that emptiness by consuming other Hearts.
Superficially, an Arrancar's Zanpakutou and a Shinigami's are the same thing, the Heart externalized and turned into a sword, but the Arrancar never imprinted their Heart onto a second soul, which never awaked its own consciousness. Thus, no sword-spirit.
Being an Arrancar is an objectively better state than being a Hollow, but the problem is that until Aizen systematized the process, the transition was extremely dangerous. At the most extreme, if the Heart fails to properly "reattach" when it is broken, then you have a naked Plus soul who has lost their innate powers and is probably emotionally messed up as hell. More commonly, you'd just lose parts of it during the transformation, diminishing your powers, also possibly with psychological consequences. If you failed to break enough of the mask, you'd have an Arrancar with a still partially-Hollow body, looking like a monstrous humanoid.
Every Arrancar can improve the power of their released state in much the same way that Shinigami can grow stronger and develop their Shikai. But Ascended Resurreccion is something else; it is essentially a singular moment in which an Arrancar seizes their reclaimed humanity and reshapes their Heart so that instead of granting them powers based on a defining aspect of their self when they were Hollow or human, it instead defines itself based on the person they are now.
Visored and Stunning Lack Of Ethics In Soul Science
Visored cannot develop naturally. They require the introduction into a soul of a secondary soul with its own Heart, which does not happen "in the wild." They're made by shoving a Hollow into somebody. In theory, this could be done with an actual Hollow with an actual personality and power-set, but in all extant cases this was done with an artificial Hollow-soul, which is why Visored don't really have unique custom Hollow powers based on their inner Hollow. It's a generic, "blank" Hollow, not unlike what the Asauchi are made of, which is why the initial transformation of all Visored have them as ragebeasts who can't even speak, unlike natural Hollows who are a twisted version of their human self and thus fully sapient, just complete assholes. The Visored's Hollows have no starting personality and thus their Heart is just "generic strong monster form." Of course, such Hollows can develop a personality over time, as is most evident in the case of White.
In canon, this happens to Ichigo because his mother had absorbed a Hollow and it was transferred to him. In Number None, this happens during Urahara's high-risk procedure to restore Ichigo's powers; the Hollowification process halfway finishes, causing part of Ichigo's soul to splinter into a "Hollow" while still remaining connected to Ichigo's soul, who due to being its own being develops a Heart of its own, separate from Ichigo's own. I use scare quotes because White has the powers of a Hollow, specifically innate access to and understanding of his Heart and the ability to shape it into a mask (and a full-body transformation if he gained total control of Ichigo's soul), but due to being plugged into Ichigo's own Heart, White does not actually have a gaping soul-hole that leaves him in constant agony and requires him to feed on other souls to feel whole again. He gets the good parts without the bad. The reason he is such an asshole is half because he spends 90% of his existence locked inside an idiot teenager's mind as he tries repeatedly and with fervor to get them both killed, and half because he is, in fact, just that much of a dick.
Fullbringers and Incoming Protagonist Nerfs
Fullbringers are formed when a foetus is exposed to Hollow energies while in its mother's womb. Hollow reiatsu is inherently corrosive and harmful to souls, and this exposure damages the overall integrity of the yet-unformed soul, so that when it becomes an actual baby and develops a Heart, that Heart has a tendency to "bleed out." In effect, what a Shinigami does through years of wielding a blank Asauchi, a Fullbringer does naturally with everything. They take any object, and cause a bit of their Heart to "flow" into that object, causing it to be connected to their soul, and allowing them to manipulate that object. If the item is of particular meaningful resonance to them, they can imprint their Heart on it in full, causing it to develop special powers that reflect their personality. Because it is a random object rather than a sword made out of monster-stuff, their powers tend to be much weirder and more varied than those of Shinigami, but because it is not a sword made out of monster-stuff, they typically never grow into the kind of martial proficiency and sheer spiritual power that Shinigami can.
In Number None, Ichigo is not a Fullbringer, because his Heart is already imprinted on a Zanpakutou, so he is "locked" into that power set.
Quincies and How To Cheat Your Way Into Even More Power
The thing about Quincies is that they are completely orthogonal to the entire rest of the power system that I just explained. Quincies are the only type of spiritually potent being (with a caveat that I will get into later) who do not gain power by manifesting or channeling their Heart. In canon, this is because their power is derived from a piece of YWHACH's soul - in effect, his Heart is massively oversized, and he takes little chips of it and buries them into people's souls, so they actually awaken a fragment of his Heart rather than their own. In Number None, it's largely a matter of trained techniques that any human can practice, plus a few secret rituals to open up your soul to the manipulation of ambiant energy. In fact, more than not caring about the Heart, they are the only type of spiritual being who does not care about spiritual power. Quincy powers rely on harnessing spiritual particles (reishi) and shaping them into weapons. A Quincy's strength is wholly unrelated to their spiritual strength, and completely dependant on their proficiency in manipulating large amounts of reishi quickly. The good part of this is that they don't have to care whether they were born with great innate potential, and the bad part is that they have a tendency to not actually train and grow their spiritual power, which means that while they may be capable of wiping out armies with a flick of their wrist (well, a nock of their bow, more like), they have the physical resilience and reflexes of a normal human being. Which was not a problem when they were having angry skirmishes with low-ranking Shinigami, but quite a problem when they went on an actual war with the upper echelons who combine quirky unique powers with baseline superhuman ability.
This is why Quincies do not have reiatsu. Even the most potent Quincies tend to simply lack personal spiritual power, and so do not emit "waste energy" as a side-effect of using their abilities.
The thing is, because a Quincy's powers are unrelated to the Heart, it is possible to be a Quincy and anything else. A Quincy Shinigami. A Quincy Arrancar. A Quincy Fullbringer. Even, potentially, a Quincy Hollow, though Hollows would generally be psychologically incapable of the kind of training and practice required to master these techniques. The reason this never actually happens is because the Quincies were a secretive cult-like organization who do not willing trade their secrets, and also because they are now nearly all dead. Mayuri's horrible experimentations on Quincies likely led to advancement in Kido and other various practices, but those were minor improvements and he never cracked the "how do I become a Quincygami" code because it turns out that vivisecting people may help you learn understand how their powers work on a medical level, but not how to train yourself into developing the same techniques. Or, in canon, because he never got a shot of YHWACH steroids.
But if you gave Ishida an Asauchi and he actually bothered to use it in battle long enough to bond with it, he could become the first Quincygami.
If I were to some day write a version of the Vanderreich arc that didn't suck, which I am unlikely to, the Sternritters would be distinguished from standard Quincies in that they found a way to channel their own Heart, which grants them unique abilities in addition to the spiritual manipulation all Quincies know how to do.
The Nemuri Project and Further Lack of Ethics in Soul Science
Mayuri's goal with the Nemuri project is to create a spiritual being who has innate access to their Heart's powers, requiring no Asauchi-bonding, Hollowification or other dangerous procedure or years-long training to make them capable combattants. Much of his work was based on Urahara's design of Project Spearhead; mod souls were fully-sapient artificial souls with customized powers, but fell short of actually having a Heart as such. Mayuri wanted to go the step beyond. The first three attempts were failures, either because the innate connection didn't work, or because being borned already plugged into their spiritual power made the artificial soul melt from the inside.
Number Four, Ururu, was an attempt at fixing this by making her Heart access an on-off switch. Mayuri shaped her Heart into a hunter-killer robot, then sealed that Heart inside an artificial human soul, and let Ururu grow into a child so she'd have a mind and body functional enough that they wouldn't melt the first time she accessed her Heart. Then, from the knowledge obtained with that project, he created Nemu, whose soul was reinforced enough to not need a switch, and who is therefore always in connection with her Heart. Nemu does not have a Zanpakutou because she does not need one, and would in fact be incapable of bonding with one due her Heart already being externalized in her own spiritual body. Then he shelved Ururu - literally, putting her in stasis in a vat - since she was only a stepping stone on the way to his successful project and not intended to serve as a soldier. At some point, Urahara managed to steal her, and the rest is history.
An unintended consequence of Nemu's nature is that since she was an artificial soul with access to her Heart from birth, that Heart never shaped itself according to her personality. It might have over the course of her life, but due to Mayuri's abusive behavior and his denying Nemu any kind of personal life or development beyond "assistant," she never developed unique powers. All her Heart does is make her generically superhuman. Which makes her fundamentally indistinguishable from...
The Heartless
A quirk of all this is that "the Heart" is not necessarily the same thing as "spiritual power." The Heart is a power; it is how an individual's core identity manifests as the potential for a unique ability. Orihime has known too much loss and grief, but she has moved on from her past and now embraces a future she believes will be all right; when that future refuses to match, she rejects that reality, and forces the world to comply. Tatsuki was manipulated, turned against the ones she loved by monsters who never even fought her, and now she will never again be denied a chance at a fair match. Chad finds in his background and the color of his skin a source of strength and validation, a left arm that can protect those weaker than him and a right arm that can smash evil. The Moth thought, if she could just fly away, all her troubles would be gone. Nemo reclaimed her humanity as she bleed out on the floor, crippled and with her soul threatening to devour itself, and will always remember that image. Yumichika takes what is ugly in the world, and not only destroys it, but turns it into a garden of beautiful flowers. Grimmjow is the primal fear of the wild. Cirucci needs dramatic lightning to accompany her fights. Ichigo will fucking cut you, don't even try him, mate. Rukia is gay. These are all the unique powers born of the most core aspects of these people's history and identity.
(The actual Heart of Rukia is the memory of her death in a blizzard, as a child trying to get back home and never making it, and the reason it manifests as a tall, beautiful woman is because she died embraced in her big sister's arms as she tried to keep her warm just a little longer.)
The Heart contains spiritual power - when one is finally in tune with one's truest self, one gains great strength from it. However an individual accesses or manifests their Heart, the simple fact of doing so makes them stronger. But the Heart is not required. Spiritual power is an organic process, it is generated in the body through the Soul Sleep and the Chain of Fate. Every spiritual being generates some spiritual power, although in the overwhelming majority of the population this is barely noticeable, and most Plus souls are so weak they don't even need food, sustaining themselves on ambient reiryoku like the autotrophs of Hueco Mundo.
So it is completely possible to develop power without ever accessing one's Heart. Before the Asauchi were invented, this was what the founding members of the Gotei 13 did - Yamamoto without a Zanpakutou had no fire powers. He was just a ridiculously powerful swordsman. Some gifted individuals are lucky enough that with training, their spiritual power allows them to match and even overcome people who actually manifest their Heart. Kenpachi has been caught in the normally brief and transitionary state in which his Zanpakutou has bonded, his Heart imprinted, and a sword-spirit begun to manifest sentience but he himself cannot yet hear it for a very, very long time. He cannot channel his Heart, whatever unique, personal power might manifest from his identity. It does not matter, because he is still stronger than most Captains on the basis of his innate strength alone. Yoruichi comes from a noble family of spies and assassins who were not yet formally integrated within the Gotei 13; she never took up an Asauchi because her family did not follow Shinigami training and practices. But because she was the scion of one of the great noble families, she was born with incredible spiritual potential, and simply learned to use it for physical enhancement and Kido. It's not that the Shihoin have their own, equally valid special techniques that make a Zanpakutou redundant, or their own way of accessing the Heart; if Yoruichi had trained with a Zanpakutou from her first years as an assassin she'd likely be even stronger than she is now. The Shihoin simply prided themselves on their independence from Shinigami rules even if they did ultimately answer to the same Central 48, which are the two main reasons Soi Fon integrated the Stealth Squad into the Second Division and all their members have Zanpakutou now. Yoruichi was simply born gifted with enormous power that rivalled actual Shinigami captains even without the power boost of channeling the Heart.
But in the end, the greatest power is only ever achieved by knowing one's truest self, confronting one's flaws and grief, and reaching not only understanding of oneself, but acceptance.