Discipline Priests struggled in WoW's early days to distinguish themselves from Holy - the more go-to, classical healing spec for Priests. Priests were (and are) the only class to have multiple healing specs, and in general, the distinction has been more one of focus than anything really radical. Holy gets more straight-up healing spells, while Disc has focused on damage absorption.
But this has been a big problem. Having a shield that prevents damage instead of a heal that retroactively gives you health back is obviously better - it essentially makes your target seem as if they have higher max health than they do.
In Cataclysm (possibly as early as Wrath, though I don't remember it then,) they started playing with an idea with Discipline called Atonement - allowing the Priest to damage enemies and thereby create additional healing effects to help one's allies.
In Legion, Discipline is diving deep into this theme, becoming almost a hybrid of a damage-dealer and a healer. Given the removal of Mistweaving's Crane Style (Fistweaving,) it's kind of surprising that they're embracing it so whole-heartedly with Discipline, but my guess is that, thanks to the existence of Holy Priests, Priest players are not totally screwed if they want to heal but Discipline winds up being a train wreck.
They're even embracing this hybridization in terms of lore and flavor. Discipline will use both Holy and Shadow spells to heal and harm.
But in terms of gameplay, they're reining in a lot of the "smart heals" that made Atonement healing perhaps too easy. You're still going to be able to damage enemies to heal your friends, but you'll need to keep close watch on those friends to make sure that they get the benefits.
A big question for the spec is what the ratio will be. After all, a healer who also does half the damage of a DPS is going to be in high demand, but a healer who does half the healing of another healer is not. In a raid, this kind of half-and-half quality actually seems really easy to use - if your raid usually runs with two or three healers, you can run with two and a half! But for dungeons, where there's just one healer in the group, will Discipline Priests find themselves benched or forced to go Holy (or Shadow and run as DPS?)
I've been adding a little caveat to all these breakdowns, but honestly, even with a new class and some totally redesigned specs, this could be the biggest example of "expect changes and iteration."
So what have we got? (Also, I think I'll include base mana percentage costs with healing specs, as that's a bit more important for them.)
Smite costs 1% mana with a 2-second cast time, dealing Holy damage to the target.
Shadow Word: Pain costs 2% mana and deals a little bit of Shadow damage up front, with a lot of additional Shadow damage over 18 seconds.
Power Word: Shield costs 3.5% mana, creating a damage absorption shield that also prevents spell pushback that lasts a maximum of 15 seconds.
Plea costs 1.2% mana, and is an instant-cast heal that heals the target for a modest amount.
Penance costs 2.5% mana and is channeled over 2 seconds, dealing heavy Holy damage to the target. This can be channeled while moving.
Mind Blast costs 4% mana and has a cast time of 1.5 seconds, dealing high Shadow damage to the target.
Shadow Mend costs 3% mana and has a 1.5 second cast time. It heals the target for a massive amount, but then deals 10% of the amount healed as Shadow damage to the target every second until the target has taken 50% of the heal from all sources (so without any other damage source, 5 seconds) or if the target is no longer in combat.
Revelation is a passive, giving Smite and Mind Blast a 30% chance to reset the cool down on Penance.
Atonement is a crucial passive, causing Plea, Power Word: Shield, and Shadow Mend to apply the Atonement buff on the target for 15 seconds. All targets with Atonement will each be healed for 50% of all spell damage you deal.
Shadowfiend is on a 3-minute cool down, summoning a fiend to attack the target for 12 seconds.
Power Word: Radiance costs 7% mana and has a 2.5-second cast time, healing the target and allies near them for a modest amount and applying Atonement to them for 50% of the normal duration.
Rapture is a 1.5 minute cool down, removing the cool down on Power Word: Shield for 8 seconds.
Pain Suppression costs 1.6% mana and has a 3-minute cool down, reducing the damage the target takes by 40% for 8 seconds. This is cartable while stunned.
Power Word: Barrier costs 6.2% mana with a 3-minute cool down, and reduces all damage taken in the targeted area by 25% for 10 seconds, also preventing spell pushback.
Mastery: Absolution increases the healing done by Atonement.
Obviously healers don't typically have a "rotation" exactly. But given that there's a big DPS component to this spec, you could almost act as if there were one. Clearly you'll want to maintain Atonement on any targets your responsible for healing. Plea is clearly the cheapest and quickest way to do this, though you'll probably want to use Power Word: Shield on cool down. Shadow Mend's damage downside of course doesn't matter much when you've got a tank taking boss-damage, as it will be overwritten before the DoT component can do much. In terms of Atonement healing, you'll probably want to maintain Shadow Word: Pain on a few targets to get constant ticking heals. Penance will provide a nice burst of healing/damage, while you can kind of titrate healing by using either Mind Blast for heavier damage or Smite for lighter damage. Power Word: Radiance is clearly meant for periods of heavy raid damage, and the cool downs have rather obvious applications.