Sauron put much of his
power in the Ring, not his soul. It's part of why the Ring grants an increasing degree of power to other wielders, starting with invisibility, longevity, spirit sight, remote sensing, aura of command, abilities not generally seen on a phylactery. (It promises infinite eventual power, but I think that's probably a lie just to tempt people.)
Sauron still didn't die after the Ring was destroyed, but suffered major power loss, even worse than when he lost it in the first place. Sauron has baseline immortality due to being an Ainu with or without the Ring.
I'd rank the Valar as "greater gods" and the Maiar as "lesser gods" in comparison to the Warhammer pantheon, both being within the wider category of Ainur. They're sort of like "archangels" and "angels" originally, Eru being the only real "God" in Middle-Earth, but Eru is also somewhat incomparable due to being the one maker of everything which is a bit above the Warhammer pantheon.
On the whole, direct power rankings are perhaps inappropriate when the Middle-Earth Legendarium is less about big warhammers and heavy metal discography faceoffs, more about poor decisions and their consequences
.
Like when the Númenoreans invade and defeat Sauron the first time, and in CK2 terms Sauron goes "Look at my Intrigue score, you should make me your Spymaster" and a bunch of the Númenoreans fall for it because Sauron's Intrigue score is in fact really good. And then Sauron persuades the Númenoreans to invade the Elvish semi-afterlife and take their immortality, and poof goes Númenor.
Or the whole Beren and Luthien story, which, while very romantic, is also a severe case of a man doing dumb things for love: "
yeah I'll go carve a Silmaril out of Morgoth's crown to prove my worth for the woman I want" and then the woman following up: "🤔hmmmm my beloved is missing, last seen sneaking up on Morgoth in pursuit of a Silmaril ... I'd better go help him".