Keeping it in the green
As a healer healthbars are just the beginning. You need to be able see who has a debuff you can dispel. You need to know when someone has a debuff that does silly damage and requires spam healing to get through. When working as part of a healing team you need to be aware of who has incoming heals and who is HoTted up. Out of combat you need to see who has lost their buff. That's a lot to keep track of on top of retaining situational awareness and staying out of trouble.
Add-ons != cheating
There are people out there that will argue that using add-ons for healing is tantamount to cheating. I wasn't healing, or even raiding, back pre-TBC, but I have been told that there were add-ons then that pretty much healed for you. You mashed your button and the add-on would pick which target most needed healing and which level of the spell to cast. Yes, that is ridiculous, but even if you wanted to, you couldn't do that now. Blizzard changed things so that kind of automation is no longer possible. Healing add-ons today are purely informational. They take the information that is there already and display it in a way that makes it easier for you to see the things you need to be aware of.
What the future holds
If you heal just fine using just the standard UI and naught else - good for you. Personally, I've used various add-ons before getting to the set-up I use today. I'm quietly pleased with my current UI (though I am constantly ogling UI threads for new ideas), but I also retain a fondness for some add-ons out there that I don't use anymore. So I have decided to do a series over the next few weeks, looking in detail at some add-ons that can be useful for healers. My aim is that it should be easy to follow even if you are new to healing or add-ons (or both). I will look at various add-ons that I currently use, have used or decided weren't quite right for me but that I know are popular - hopefully you might find some add-ons that are right for you.
A post on Grid will be coming this way in the next couple of days.