/scoff at the Mind Control comment. That's something people are trying to propagate because they hate it when it happens. It's harder to do than it once was, because everyone's trinket allows them to escape now, but it's not nothing.
Priests do not hold the exalted position they once did. Other class' healing abilities have gained more prominence in raiding environments. All things being equal, every healing class can carry its weight in a raid healing environment. Thus, raid slots are allotted (again, all other things being equal) based on what additional utility they bring. Currently, the other classes top priests in that department. That is, an additional priest adds no additional utility, whereas another paladin brings another raid-wide blessing, another druid means another battle rez, and another shaman brings another set of totems. Thus, although one or two healing priests are guaranteed raid spots, additional priests are hard to justify when other classes are available.
All of that said, there are never enough healers. People take what they can get. It's not been a problem for me.
Shadowpriests are more in demand for raiding, due to the fact that they serve a vital mana-battery role in addition to doing hefty damage.
In arenas, priests have significant problems, but so do paladins. The problem with priests is that, at low levels, they are incredibly vulnerable in comparison to paladins. A priest with low stamina and resilience is so much dead meat. It takes quite a bit of gear to get priests' survivability up. Once it does, the advantage of priests becomes clear: the ability to grant considerable survivability using instant cast spells alone. Paladins can't match it, which is a problem.
Shadow priests, in contrast, have struggled in arenas of late. The reasons are two-fold. First, for a class that relies heavily on DoTs, priests have nothing like Unstable Affliction (or the sheer crowd control) that warlocks have to allow their DoTs to tick. Additionally, the anchor of the shadowpriest ******** Mind Flay, roots the shadowpriest, which is quite a liability.
(Don't take the mage comparison that seriously. Although three-minute-mages have a certain notoriety, the most valuable things a mage brings to the arena table are counterspell and polymorph.)
HAVING SAID ALL OF THAT, skill > all. Not all things are equal. If you're a good priest, you will be treated and appreciated as such.