All Priest types heal, buff, debuff and resurrect, they each heal in a different way and have a different Utility factor.
Druids :
Both wear leather armour and both used HoT (Health over Time) as their specialist heal. They both use fire and ice based nukes and DoTs. They both get to use the Druid Portals and run speed buff. This is where the similarities end.
Fury has strong nukes and a group invis and its secondary and tertiary heal lines are direct heals and delayed group heals. Fury has good buff for caster classes. Fury is generally regarded as a good soloer due to fast kills. Fury gets a group invisibility spell which is very useful. Furies chapchange to a Lion form which adds to thier dps.
Warden has more heal lines and weaker nukes. All warden heal lines have a HoT element and they can all be stacked up. Wardens kick out more replenishment heal than any other priest class although this does not make them the best healers, as such. Warden can get a badly damaged group vack on its feet more efficicntly than any other. Warden gets very strong root spells and a group Evacuation spell. Wardens buffs are useful but not amazing and they are more melee oriented than the furies. wardens change to a wold form which adds to their defenses.
Clerics
Clerics are more technical than other healers as they rely on reactive heals and proc (lotto) heals which have a chance to go off and these are in the form of debuffs to the enemy. The straight reactives heal any and all damage a set amount of times. Generally they are a powerful healer but with slow spells, a lot of timing and intelligent casting orders are needed to make the most of the class. A lot of players get to grips with this class until critical situations then they fall to bits. These days, there are porbaly more baandoned clerics than any other class in the game as new players roll something easier. Clerics wear plate armour so can take some hits.
Templars can only be played by good aligned characters. They have a huge range of powerful heals and reactives. their direct heals are the strongest in the game but SLOW to cas and power hungry. By combining debuff, looto and reactive heals, templars conserve their power and throw out the directs only when necessary. Templars get some quite strong nukes as well, especially against undead. Templars a slow to solo although the achivement skills can get past their main problems of slow casting and lots of interruptions, I personally built mine for solo early on and respecced for free later once I started grouping regularly. It is only at higher levels that the boosts to heals and buffs are really essential and best to spec for solo early on to have a better game experience. I really like this class, it is technical, challenging and extremely rewarding. (its not my main btw) Templar has some great buffs for Tanks too !
Inquisitors have similar heal methods to the Templar but they have more lines of DoTs and debuffs, some of their debuffs are very powerful. the solo much more readily and can even do a spot of Tanking at low or mid levels if they build for dps. Inquisitors are the evil counterpart of the Templar and dont have the same raw heal power but are still (as all priest are) an effective and capable healer when played well. Inquisitors debuffs are varied and useful, they can debuff any type of mob to reduce its primary stats. Inquisitors buffs range from health and mitigation to damage boosts and damage procs. whereas Templar adds to defenses generally, Inquisitor boosts damage output generally.
Shaman
Both Shaman types use powerful wards to prevent damage altogether snd have some strong direct heals too although these are slow to cast. Shamans use chain armour so are fairly resilient and have a mix of nukes, debuffs and DoTs.
Mystics are the more defensive and can shapechange to Bear form which boosts health, power and defense. They amongst the best melee dps of the Priest classes but are by no means strong in this area. Mystics Wards are their main line of healing, the strongest being the group ward which is a clooective ward so if only one person takes damage they get the full ward potential which gives time to lay down a singke ward. Group wards can draw a lot of aggro and both types of Shaman need to learn to deal with the hate they generate. Mystic buffs affect power pool and resistance to disease and poison. Theiy ahve direc, DoT and damage + debuff spells based on either disease or cold. Mystics can also debuff, slowing down the opponents. Like Templars this class has strong direct heals but these are slow and power hungry and are their to fill in the gaps that the wards do not cover. Its nice for mystics to have some backup such as a Drui for some spot healing and extra nuking but they can heal fine solo too but its a lot of hard work.
Defilers are a very offensive version and very evil with it. They have a slew of powerful stuns and disease or posion based DoTs. Their buffs are more offenisve as well with boosts to strenght and attack speed. Other debuffs decrease posion and diseas resistance making the dmage spells more effective. Defilers direct heals use up some of their own health, not a lot but become impossible to cast when at very low health. The Defiler also gets a weak pet but this can be buffed and made fairly useful although like all non-specialised pets is vulnerable to AoE attacks. Even the deiflers wards have bite and act as a weak damage shield as well. As with Mystic the stringest Wards are collective group wards which absorb a large amount of damage if only the Tank is getting hit but get blown away fast if the group is all taking damage. Again, backup is nice for fast health replenishment but a hardworking Defiler can also solo heal a group. Defiler soloes faster than Mystic.
Phew ! Sorry for the long post and I am sure I have missed some stuff but I hope that gives a clear idea, at least for general grouping. Don't get put off by anything you read about solow soloing, low-mid level builds can get past the problems some defensive classes have. My main is a Warden and I solo just fine thanks.