I play all three, and since you've already gotten some more in depth answers I will just give a quick comparison

AST- Fast big heals, and some awesome buffs. The cards are random but that's part of the fun to me. Synastry is a huge life saver and Lightspeed is amazing to pop out some quick heals, or dps!

SCH- Medium heals and Shields. Lots of fun skills here with multiple shielding abilities and some large instant heals.The fairy is amazing. You can choose Eos for more healing or Selene to buff the dps.

WHM- Slow but potentially huge heals. There is of course Presence of Mind to help speed things up, and a couple of fancy things to instant cast, but even so it still feels overall slower than the other healers. WHM also requires the most mp management.

This is just from my perspective and playstyle. Most importantly they are better together. Where one lacks the other shines and when you are in party together you can feel the jive of having different types of heals going on