+Damage and Healing works with all your healing spells (Healing Touch, Lifebloom, etc.) as well as with all of your damaging spells (Moonfire, Wrath, etc.)
+Healing works with only your healing spells and does not work at all with your damaging spells. This is being changed in the next patch though, so most items with +Healing will give approximately 1/3 of the +Healing bonus to +Damage and Healing, to give Resto Druids and Shamans, Holy Priests and Paladins to ability solo semi-decently without having to respec.
For your healing set, you want +Healing, for your casting set, you want +Damage (along with Intellect, mp5, Stamina for both and Spirit for healing). How much of the +Healing or +Damage your spells get depends on their casting time and talents. WoWWiki has a nice
page on how much each spell gets.