Shadow Priests have always hinted at being connected to the same powers as the Eldritch Old Gods, and things like Void Tendrils or really just the fact that they assault their enemies' minds has played into that. That theme of craziness at the center of their gameplay is getting put front and center, with new resource that replaces Mana called Insanity. There are also no longer any Shadow Orbs.
Insanity is actually pretty odd as a resource, as you have a lot more control over how you generate it than how you spend it. Your spells will generate different amounts of Insanity until you hit 100. Then, you will automatically shift into Void Form (Shadowform will be always-on while you're specced Shadow.) Void Form reduces the cooldown of Mind Blast by 3 seconds (to 6) and increases all Shadow damage you deal by 30%.
While in Void Form, you'll get 1% Haste every second, and this buff will last for 20 seconds after Void Form ends.
While in Void Form, you'll lose Insanity at a faster and faster rate over time, until it becomes impossible to maintain it and you'll drop back down into the old Shadowform, but of course, with that 20 second buff to your haste. You'll still be able to generate Insanity while in Void Form, but it will drain faster and faster.
Many of your spells should look pretty familiar.
Shadow Word: Pain remains an 18-second DOT the does Shadow damage.
Mind Flay remains your channeled filler spell, dealing Shadow damage and generating 10.02 (weird number) Insanity on its full duration.
Mind Blast looks largely the same as well, with a 9 second cooldown and a 1.5 second cast time, dealing a bunch of Shadow damage and then generating 20 Insanity.
Void Bolt is a new spell, instant on a 6-second cooldown that deals shadow damage to the target and extends the duration of Shadow Word: Pain and Vampiric Touch on the target by 3 seconds. This can only be cast in Void Form.
Vampiric Touch is slightly different, a DoT that heals you for all the damage it deals, with a 1.5 second cast time and a 24-second duration. If dispelled, the dispeller flees in horror for 3 seconds.
Shadow Mend is a heal that costs 10 Insanity if you have it (though if you have none, it costs nothing,) healing the target for a large amount but then placing a DoT on you that deal half the damage it had healed for over five seconds, though if you leave combat, the DoT will end (and presumably not even start if you're already out of combat when you cast it.)
Shadow Word: Death is instant on an 8-second cooldown, only usable on targets below 20%, and does a bunch of Shadow damage. If you kill the target with it, you gain 30 Insanity.
Shadowy Apparitions remains a passive that causes you Shadow Word: Pain crits to summon an apparition that flies toward the enemy and deals shadow damage.
Shadow Fiend is a 3-min cooldown that summons a... wait for it... shadowy fiend to attack your target for 12 seconds.
Mind Sear remains as a channeled AoE and seems more or less unchanged, but it does not appear to generate any Insanity.
There are a number of utility spells as well, but that's it for rotational ones.
So, before talents or artifact traits, the rotation seems not too different to where we were before:
1. Keep up Shadow Word: Pain
2. Keep up Vampiric Touch
3. Mind Blast on cooldown.
4. Mind Flay to fill gaps.
When you're Insane, you can alter it a bit, using Void Bolt about once every three seconds to keep the DoTs running and then try to keep generating as much Insanity as you can to get that buff to stack up. I don't know that there's even a cap on the buff.
While it's certainly different, given the lack of Devouring Plague, most of the bones of the rotation seem pretty untouched.