Priest is the "pure" healer class and Mystic is a hybrid. With Priest almost every spell you get consists of some type or variation of single target heal/group heal/regen and all you really do is spam them and buff your group, dispel hostile debuffs, and debuff bosses. You get a couple of single target nukes, a single target sleep/cc ability, and I think a small melee AoE and maybe a DoT. They'll get you through solo leveling easily but in groups and PvP all you're really going to be doing is running around mashing your heal keys with the occasional rez and AoE mana replenishment thrown in.
I haven't had much experience with a Mystic but one of their primary heals consists of dropping health and mana globes that people have to run over to and pick up. I think they get a couple other different heals later on that work pretty good but have longer CD's than priest. They get a pet and they can do more damage.
The Tera community has always bitched about Priest vs. Mystic and how good their heals are compared to each other, but I've always found Priest far and away the best pure healer that's practically idiot proof while Mystic takes a bit of finesse. Before I took a break most groups preferred Priest over Mystic too, but to be honest with the addition of the laser tits class shit tends to die so fast heals are hardly needed with most instances till possibly end game.
Mystic definitely seems to be more battle fun, but Priest can carry a group of retards while masturbating most of the time.
Edit: The Regens mark the ground with an AoE and everyone in them gets healed, while the group and other heals requires you to "activate" the spell by using it, using your mouse to highlight the other players running around (once you mouse over them they get "locked in" until they run out of range), and then you press the skill again to cast the heal that hits everyone locked in. You'll have to do this entire process each time you cast the ability. I think you can heal everyone but one or two with this spell so it's not a true group heal, but with it being spammable and the large amount it heals for you'll be able to switch people in and out fast enough. It's really strange at first to get used to and the only issue I have with it is that every now and then you'll get a low hp class that stands in front of a boss and stacks on top of the tank making it really hard to target one or both of them, but this usually only happens in really bad PUGs and by end game is mostly non-existent.