Reputations concern your name to the rest of the Imperium, even ones who have not met you personally. A Reputation is a First Impression: Having a Good Reputation with the Ordo Xenos will mean that they will think more highly of you, whether because you've worked with them before or have made a name smiting the alien enemies of the Emperor, and so they will probably come by more often if you ask for their help. Conversely, a Bad Reputation with the Ordo Xenos will make them less likely to help you. If your rep gets bad enough or if the evidence is damning enough, they might even move against you.
None of this makes it impossible to cooperate or conflict with the Ordo Xenos.
The Ordo Xenos won't simply hand you Deathwatch marines or pieces of their ridiculously overpowered wargear as and when they like, because they need that wargear to shoot that Hive Tyrant in the face. Instead, what it means is that they will be more inclined as a whole to support you and listen when you come calling. Need someone to back you up at a tribunal? Call the Xenos. Need the Deathwatch? Call the Xenos. Have to root out alien taint on a planet or (Emperor forbid) within your own Chapter and either have not the resources or the time to do it? Call the fucking Xenos. What they do then is up to them, but they'll show up.
Meanwhile, Favors mean another thing entirely. These are intangible things, the sum total of blackmail, goodwill and Emperor-given obligation that you have over a specific unit or individual. You will never have favors from the entirety of the Ordo Xenos, for instance, but you may have favors from Inquisitor Eldarshanker. This manifests in three different flavours: Minor, Major, and Honorbound.
Minor favors are... well, not exactly minor, but they are relatively minor incentives and such, to cut through bureaucracy or push an organisation/individual into a certain decision they are on the fence on. Say you helped Inquisitor Eldarshanker take out a Farseer leading an Eldar Warband somewhere, leading some ridiculously convoluted plan to buy their Craftworld a couple more hours via diverting a Waaaagh into a Hive World or something. He'll owe you a Minor favor, to be paid elsewhere.
Now, say you need that favor to call upon Ordo Xenos assets again when, say, you need to virus bomb because the planet is literally made of Orks. The Ordo Xenos was going to do that anyway, but it would take way too long, like months or even years. You then call upon Inquisitor Eldarshanker, cash in your favor, and cut through that bureaucratic inertia with the power of blackmail friendship. The virus bombing now happens within weeks, because the Inquisitor decided to do it himself. He was in the area, he has other things to do, but why not?
Major favors are a bit more substantial. A Major favor is asking for an investment of resources that the organisation/individual was not planning to do but is still within their purview. Using the Ordo Xenos again, this would be like requesting they deploy a squad of the Deathwatch to extract a bunch of neophytes on a planet overrun by Tyranids[1]. This falls under their purview (cockblocking the alien), but they weren't planning on that. Maybe that squad was going to assist Inquisitor Eldarshanker on clearing out a coven of Howling Banshees. Maybe they were going to assist a Crusade into Tau territory. You don't know, but they have to divert assets for the explicit purpose of aiding you.
[1]Why they were on that planet, I don't know. Training, rites, the point is you have a bunch of neophytes about to get their face raped by Tyranids of all shapes and sizes.
Honorbound favors are the kingpin of favors, century-old, Primarch-mandated demands. At this point, you're asking for something insane and might be slightly heretical (but you're doing it for the Emperor so it's totally not (probably)). Using the Adeptus Mechanicus[2] for example, this would be akin to asking Magos Biologis Love&Stuff to personally generate and clone new gene seed to create new marines (though obviously it is in case you guys lose horrible amounts of brothers in the Crusade you're in (what Crusade? Shut your mouth)), and they owe you because you were the guys that saved one of their God-Machine superTitans.
The Codex Astartes does not condone this course of action, it's heresy beyond heresy to generate more gene seed than what is provided by your brothers via the Holy Progenoid Glands, and the Inquisition will be glaring down your neck because 'Why do you need so much gene seed, hmmmm? Why did you ask the Mechanicus? Is it because you're TAINTED?!', which will probably result in you being declared Excommunate Traitoris and bring unending shame onto the Blood Angels for choosing you of all people to lead a Space Marine Chapter, even if you had completely legitimate reasons for doing so.
[2]Shut up couldn't think of Ordo Xenos examples
The degrees of Reputation you can earn range from Despised, to Hated, to Bad, to Suspect, to Neutral, to Good, to Liked, to Honored, to Exalted.
Despised (you will be seen as one step short of a heretic at all times)
Hated (you will be watched and all attempts will be made to obstruct you)
Bad (they might still work with you but it's certainly not a priority)
Suspect (You smell fishy, so they'll be keeping an eye on you)
Neutral (They'll jump at the chance to fight alongside you, but have no strong opinions one way or another)
Good (they like you and might vouch for you if it's not too inconvenient)
Liked (need help? You know who to call)
Honored (You are nothing less than the ideal of a Space Marine, spoken in the same breath as the Chapters of the First Founding)
Exalted (nothing you do could possibly be for anything but the Emperor himself)
