I was gonna mention Hermetics in Early- (and just the more pre-1066 start) game too first of all. They are available for many religions and will boost tech research more than anything else can.
Furthermore they let you create artifacts, and certain really nice things that only take full effect in very specific circumstances-
But if you happen to be building a great work in your Capital, and have the Blast Furnace, and combine it with a carpenters guild from business focus, you don't even need architect anymore to get stuff done affordable and in , comparably, no time. Thanks to this combo, I recently was able to successully construct a harbor in a game as Navarra, earlier than I was able to build wonders when playing "Big Shots" usually- I had had build the quarry too, so I gobbled up a whopping 50% building cost discount from province modifiers easily, and affecting the building time in a similar manner, thus the Architect managed to complete Stage 1 in his lifetime, and his daughter raise it all the way to full tier 4 during a 30 year reign.
Warrior Societies, however, arguably have much stronger perks and features, making your character more battle-resilient, and allowing to create additional bloodlines to pass on to heirs. It is really under any circumstances worth to briefly switch to a pagan religion and join a warrior club at the start of a rule, and then re-convert.
If you use secret conversion, this can be done without any additional unwarranted effects such as moral authority or prestige loss, for a moderate amount of 500 piety- nothing hard to achieve in quick time specially for bigger realms.
Satanists can in theory be used any time. As possessed courtiers and vassals are prone to get the idea murdering you might rank them up in the society, it's no downside to not be able to use this power on Vassal Kings- can always plot to murder and use the society support, or strike them with the plague, after all!
Much more important and powerful are the manipulation powers of the satanists, specially once you reached the top tier:
"Society Members agree to diplomatic demands" is really the most underrated tooltip in the game- you recruit your neighbour to the society, declare war on him, and he will right away accept your suggestion to enforce the demands.
Last but not least, it does let you systematically breed satan spawn, which is quite powerful if there's not eight older brothers in line and a girl starts to murder everyone with agnatic inheritance. Downside, no artifacts to gather as much as elsewhere, and of course it might backfire a lot and hard.
Monastic societies have their justification in any religion- while the perks seem less great (lower fertility, most notably), they do help you to become a better person and ruler.
Thus, while the other societies are a bit of a wager, as membership does not guarantee to be fruitful (warrior quest might fail, hermetic gathering too), membership in a monastic society can really barely anytime hurt.
In particular you can observe these benefits on the AI easily- monastic society AI rulers will perform significantly better in average than demon worshippers or others will, for the simple reason they tend to gather the one or other "nice" trait, ensuring their vassals like them and abstain from revolting.
With Monastic Society membership, even AI controlled Michael VII Doukas can turn out to become a competent and able Basileus that holds his realm together.
Thus, they're best applied when you take over a character from whom you cannot expect great things in other areas (as he lacks intrigue or learning), and has no option to join a warrior society. Taking over as older characters might be worth it as well, as you can already do really neat stuff (notably, passing good traits to kinsmen and vassals) on tier 3, opposed to the other societies that do require top tier for the really good stuff.
Last but not least the assassins.
Limited more in access than all others, hence why there's not many situations to consider them to begin with. They can be rated as specially useful for
-Any Shia ruler
-Any ruler early on, specially small realms, as long as the character has decent intrigue- if you make it to grandmaster, you can spawn event troops that might massively enhance your conquest potential all of a sudden
They're generally worth considering in later game too as they do give decent stat bonuses, can teach you positive traits (rarer than the monastic society, though), and also have two perks of the satanists, without the constant risk of growing clubfeet: For once, worldwide society murder plots (asking society members to aid in plots anywhere), and the grandmaster's ability to enforce diplomatic demands.
While the later is slightly less useful than with the satanists, as you cannot recruit as freely for the assassins (You can, but characters being sunni or christian will almost always decline) and thus won't as easily trick your neighbours into folding to any war declaration of yours, it can still be of use (just keep trying to recruit that neighbour Sultan of yours, or his heir and then murder him with massive society plot power).
Downside: No specific artifacts to be created from this society.
