The release of the 5.2 PTR is pretty close now, and I thought I'd take a moment and list Priest issues and flaws in both aspects of the game. Since it has been known for Blizzard to read, and introduce, feedback from sites like these, maybe some good with come of it.
In PvE is very strong, bordering on being the "best" healer. It probably could use a slight nerfs in a couple places. Temporary spirit procs, such as tailoring, trinkets, and Mana tide are boosting regen levels far past other healers when combined with Rapture. GC has already hinted this is going to be removed in 5.2. It is also possible DA(or Spirit Shell) will be lowered to help bring the healers into line.
In PvP, Discipline is struggling. Mana continues to be a considerable problem for both healing priest classes. Mass Dispel is prohibitively expensive to use. Lack of CC, except for Fear (which some classes can negate even without the use of their trinkets) is probably the most severe problem. Survivability is also questionable; Spectral Guise is now nerfed into the ground, Inner Will is a paltry 2% movement speed at the expense of a ton of mitigation. Blizzard has acknowledged Discipline needs intervention in PvP, but no words on what they will do, except they will try some things on the 5.2 PTR. It has been widely suggested Glyph of Desperation will become baseline.
Holy has a variety of issues preventing it from being a competitive spec. Throughput is very low, especially when compared to Discipline. Mana regen is also noticeably worse then other healers. It is possible Holy Concentration will be reintroduced soon. Going forward, GC has noted Chakra's passives will be removed (probably no earlier then next expansion) and stances will be purely mechanical differences, such as the behavior of Chastise and things like Renew refreshing.
In PvP, Holy is a complete write off. It suffers from the exact same things as Discipline, as well as missing things like Focused Will (which will likely be extended to Holy) and glyph of Inner Focus.
In PvE, Shadow is doing okay. While DPS right now is fine, the scaling problems that has plagued the spec for years are starting to re-appear. DoT's, such as Vampiric Touch, Shadow Word: Pain and Mindflay are scaling nicely. Resource generation is mostly the problem here. Mindblast/SW:Death doesn't scale with Mastery, and scales very little with Haste. Devouring Plague also suffers from questionable scaling. These issues will continue to become more severe as gear progresses.
Shadow in PvP is doing well, and is one of the more powerful specs although not nearly as powerful as it was. Although Shadow's healing potential was lessened considerably, no one really disagrees with the need for the change. The spec still suffers from being pressured; to the point it's damage is almost negated entirely. Personally, I'd like to see Mind Spike not remove DoTs so at least something may be cast.