This is coming from the point of view of someone who's currently got this healing arsenal: 60 AST, 56 SCH, 50 WHM... Also bear in mind that I'm a scrub casual that enjoys the abuse of DF as my main form of playing.

AST
I like AST a lot. I chose it mainly from a "lore" perspective, as my character is sailor moon based, so I thought "oooh stars," and figured she had to be one, lol. But I do quite enjoy it. The RNG of the cards sometimes makes me laugh. There's nothing quite like setting up an AOE-Bole or AOE-Balance in a fight (which then promptly misses the BLM EVERY. SINGLE. TIME.)... but sometimes you burn all your cooldowns and all you get is a buffed Spear which makes me want to hang my head in shame.

Heals themselves feel powerful enough to keep up. Sometimes, between all the dodging and healing, it gets tricky trying to weave card buffing in there too.


SCH
Currently enjoying it a lot. Haven't seen any new healing spells yet by 56 (haven't done the job quest yet). It's feeling a bit under-powered to where I was as an AST at this point of the game, actually. Can't tell if it's because I've been splitting my attributes between SCH and SUM. It isn't gear, as I saved it all from when I was leveling my AST up.

I like SCH for the proactiveness and being on top of incoming dmg. I usually find it better to have the fairy on obey, otherwise she just burns all her abilities whenever. "Wow, I could really have used whispering dawn right now... too bad she popped it during that single target pull where no one took damage."

The challenge this class brings is being aware of your aether stacks and not burning them out too fast.


WHM
It's been a long time since I've played my WHM, so I'm super rusty. But way back when, I enjoyed it because of the over-powered heals. Everything you have in your arsenal is geared towards recovering from bad pulls, heavy pulls, mitigating spike damage reactively, etc.

Challenge: This is probably inexperience talking but compared to the other two classes, I felt that this one has the most potential to burn through mana. Shroud is your regen spell (it also reduces your aggro, so it's very handy), but the last time I played this class, it didn't feel like enough. Aggro generation in general seemed higher as a WHM compared to the other two. Don't cast renew habitually. If it's still up on the tank before the next pull, s/he will never clear it off and you will immediately gain 100% of the aggro off the pull.