Paladins may be only about healing (I'm talking holy, not lolprot), but that means that they must really excel at it. They need to avoid CC chains, learn to keep up their mana, have good understanding with their dps when they have to back out and go defensive so you can catch up on heals. You also have to make the most of your HOJ and whatever small offensive ability you have. If you have to waste alot of mana unnecessarily because of not paying attention or something, then it's tough getting it back. Keeping up with healing the whole match can be undone by a bit of burst, and there's no lucky burst or cc chains for you like other healers can accomplish. Good pallies also use their abilities to the max, like switching auras/seals/and judging often.
There may be alot of oh crap buttons like bubble, but it's needed when you think about how easy it is to shut down a pally sometimes.
Priests are the opposite, it's rare that you can play defensively a whole match and win. You'll go oom way before the other healer. So not only is it necessary to heal, avoid cc, and other main healing oriented things, but you have to cast some damage attacks, mana burn, dispel, and use effective fears each match to stand a chance. It's easier to burst heal and keep up mitigation, but actual consistent nonstop healing of people is rough and probably can't be done for very long.
Shamans are similar to priests as they'll have to do some offensive moves to even come close to a victory, since their mana regen isn't that awesome either. I'd say they have the highest skill cap of the healers. Some shamans can negate a good portion of casts, as well as kite melee. Right now they're a bit weak, but with buffs, I imagine shamans being quite represented.
Druids are a mix of both. In BC they were just invincible offensive and defensive machines. Right now they're kind of like paladins in the way that they win just by keeping a steady stream of heals. Although, it's alot easier to do it as a druid. Alot of CC's can be negated as well, not to mention there's always the option for having the most mobility of the healers. I'm not calling druids skilless, even if they have an easier time healing, there's still things that can randomly gib them or their partner that they'll have to learn tricks for. It's not like they're that weak against burst anymore though, swiftmend and nourish take care of any problem. They are awfully mana efficient, and can reset the fight alot easier than a holy paladin with travel form, cyclone, and roots.
Not to mention there's still a good deal of offensive ability. I'm glad when I see a druid sticking to tree form and just healing. At least there's the possibility of him getting behind on heals eventually. With a druid who nonstop cyclones, roots, and puts up dots, he adds an element to the fight that you have to work around. With a good partner, he'll drink easily with no problem, and your team will die much sooner.
If you want to roll a healing class, try a druid. Easy enough learning curve to survive and win more games than other healing classes in the same learning time, but has a high enough skill cap.
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