So first the big caveat I always feel I should start with when I talk about Priests. Of the eleven classes in WoW, Priests are the one I have played the least. I got mine to level 100 and then more or less dropped him. But I did get him to 100, so I have some sense of how the class works.
Shadow is one of the specs that has gotten a new resource system, and the spec's connection to the Old Gods, which has only been strongly implied before now, has become basically explicit. So let's get into things.
Insane in the Membrane:
Much like Balance Druids and DPS Shamans, Shadow Priests still have a mana bar, but it has been bumped down under the main resource bar to make way for Insanity. Yes, Shadow Priests channel their own craziness to attack their foes. Insanity is a very unusual resource in that almost all of your abilities generate it, rather than spending it. In a sense, it works a little like Demonic Fury did for Demonology Warlocks in Mists and Warlords, but not exactly. We'll get into its mechanics in a moment. It's represented by a bluish-lavender bar under your health bar.
Shadowform Gone/Always On:
Shadow Priests will now always be in Shadowform, meaning that in a sense, the ability no longer exists. However, cosmetically, Shadow Priests will be darkened and translucent at all times. I also believe that the more Insanity you have, the darker the effect will be.
Keeping the flavor of being unable to cast Light-based spells, you now have a new heal called Shadow Mend (something we've been seeing enemy NPCs cast since vanilla,) which heals for a large amount and then, if you're in combat, puts a DoT on you until you've taken half of the heal's amount in damage (from any source - the DoT itself or external damage.) Outside of combat, this is just a straight heal, and there's a talent early on that changes the DoT component to a HoT. You also have Power Word: Shield, which counts as an in-between, Discipline-type spell.
Core Rotation Looks Similar:
Many of your core spells are going to be pretty much the same. You'll still be DoT-ing up your targets with Shadow Word: Pain and Vampiric Touch, and you'll still be hitting them with Mind Blast and filling in with Mind Flay (which can be replaced with Mind Spike with a talent at level 100.) All of these abilities generate Insanity, with the DoTs doing it on cast and every time they tick. Your little Shadowfiend summon is still a DPS cooldown, and it will now generate Insanity for you each time it hits the target.
Goodbye, Shadow Orbs, Hello, Void Form:
Shadow Orbs are gone, as is Devouring Plague. You're no longer focused on building to this major DoT. Instead, you'll try to get your Insanity to 100, at which point you'll hit Void Eruption. Void Eruption deals damage to any targets affected by your DoTs and then puts you in Void Form.
Void Form will begin to consume your Insanity. It starts off slow, but the longer you're in the form, the faster it will drain away the resource. The more spells you're able to cast and the more DoTs you're able to maintain, the more Insanity you'll generate to maintain the form. When your Insanity hits zero, you'll exit the form.
Void Form does a couple things. First, it increases your damage dealt by 20%. Your Mind Blast also has its cooldown reduced by 3 seconds. You gain 1% Haste every second while it's active, and after Void Form ends, you retain that stacked haste buff for one minute.
Finally, Void Form transforms Void Eruption into Void Bolt, which is instant-cast, dealing damage to the target and refreshing the duration of your two major DoTs.
The New Shadow:
So while it's not as radically different as, say, Survival Hunters, Shadow does work pretty differently. You're going to want to maximize Insanity generation both in and out of Void Form, but particularly within it, to try to get that Lingering Insanity haste buff to stack as high as possible. If you manage to stay in Void Form long enough (I want to say 15 haste stacks) you'll even start to sprout tentacles, which is pretty awesome/freaky.
So that's the basic gist of the changes to Shadow. And as we say in the Void, Iä! Iä! N'Zoth fhtagn! Ph'nglui mglw'nfah N'zoth Ny'alotha wgah'nagl fhtagn!
EDIT: I also feel I need to draw attention to the most ridiculous talent in the entire game: Surrender to Madness. This is a highly impractical, but potentially powerful niche ability that is the only ability in the game (now that Divine Intervention has been gone for a while) that actually kills you.
The ability has a 10-minute cooldown (I'm currently waiting on its cooldown to switch to a different talent on the PTR) and causes your Insanity-generating abilities to generate 200% more Insanity (so three times their normal amount.) It also allows you to cast while moving until you exit Void Form (I actually don't know what happens if you cast it before using Void Form, or what happens if you never enter Void Form while it's active - though if you didn't, what would be the point of the ability?) after which you die. Horribly.
Dying Horribly means that you can not be battle-rezzed. You get this insane burst of damage and you can maintain the Void Form buff for a good long time (I had like four or five tentacles waving around me by the end) but you better be near the end of the fight, because you are not getting back into it. The debuff on you when you're dead lasts a couple minutes, so theoretically if you used this early on, you might be able to get rezzed later on, but I can't imagine being dead for several minutes of the fight is a DPS gain. I think the only really practical time to use this is if you have a really tough burn phase at the end of a boss fight - something like Professor Putricide, where you just need to pour out as much damage as you possibly can or the raid will get overwhelmed. During that period, a Shadow Priest who surrenders to madness might be able to very significantly increase the damage he or she causes, and once the boss is dead, your buddies can just rez you when it's time to distribute loot.