Well to compare them, this is somewhat in depth.
Priest has a lock on heal, a small AoE burst heal, a burst AoE lock on heal, two AoE heal over time; Mystic has a lock on heal, a totem AoE heal over time, can drop motes that heal over time, a straight line boomeran heal, and a pet that heals the caster and party members over time. In sum Priest heals are stronger for bursts and Mystic is more sustained.
Priest has to cast the buffing skill every 20 seconds to keep the party buffed with power and attack speed, while giving two buffs that increase MP/HP regen, power, and endurance before the fight starts, this ones last for 15 minutes and are lost upon death, so the Priest has to rebuff after resurrecting someone (if the priest dies the buffs are still on the other players); Mystic has auras, they give attack speed, power, MP, and crit, they also cast a 10 minute endurance buff before starting the boss (if the M dies no one gets the buff until the M is back up with auras on). Side note: in the console version this numbers may vary, making the Priest's buff give more power than Mystic's.
Priest debuffs for 10% endurance reduction for 11 seconds with a 4 seconds cooldown, so it must be refreshed often; Mystic has a 9% endurance reduction (becomes 10% later on with patches) that last for 17 seconds with 4 seconds cooldown, they also have an extra 5% for 25 seconds but with a higher cooldwon, the last one is only used during burns. This is all debuffing the boss. Side note: Mystic doesn't enter combat while debuffing, so it moves faster than Priest.
Priest gives mana through mana charge, it charges and on release replenishes mana, the more it charges the more mana it gives up to a cap, the earlier buff, and one of the AoE zones that can be glyphed for extra mana; Mystics can drop motes that give mana, the aura gives mana, and has a two part skill that needs to hit a target to charge up, and on release it replenishes mana. Side note: Priest is safer on this but gives less mana, also Mystic's skill can be glyphed for 50% damage reduction.
Priest has a big AoE cleanse; Mystic has a lock on cleanse, the motes they drop cleanse and the pet has an AoE cleanse.
Priest has a lock on regress; Mystic has an AoE regress around the caster. Side note: Priest is safer for regressing as it doesn't require to be at melee range from the boss.
Priest has two mobility skills, one of them is an iframe; Mystic has one mobility skill and it's an iframe. Side note: Priest's skills travel for a shorter distance.
Priest has a strong self heal, can walk into the AoE lock on heal, and into the AoE heal over time area; Mystic has it's pet, and the motes can be glyphed to restore 20% HP to the caster. Side note: Mystic's pet cooldown is high enough to make it slightly unreliable, unlike Priests that can heal up for a larger amount.
Priest has a shield that covers 38k called Kaia's shield; Mystic has a 8k shield which comes with the totem. Side note: Kaia's shield also prevents knockdown and push backs.
Priest can pull players and make them stand up if they get knocked down; Mystic can't do either of these.
Priest has a 20 minute self resurrection skill with a 8 (?) minute cooldown that can only use on himself; Mystic has the same but requires a party member to be used, as it locks onto someone and buffs that someone and the caster, this will be nerfed later on.
Priest resurrection skill is faster than Mystic with less animation lock.
Healers damage doesn't matter but Mystic wins over Priest. Also Priest is prefered in raiding but Mystic in 5 man, either class is good and strong so whatever you pick you'll be fine, altho people like Mystic more since it becomes a flat out a better healer than Priest later down the road. Sorry for such a wall of text.