Healing is the role where add-ons change the most. You stare at raid frames, not the game world.
Raid frames
VuhDo: frames plus mouse-button healing in one add-on. The pick while 
Clique has no Forever version.- Grid2: lighter frames; pair it with your own keybinds.
Cell is popular on retail but has no Forever version yet.
Around the frames
Deadly Boss Mods (DBM) or 
BigWigs Bossmods for dungeon and raid warnings.- Details! Damage Meter: its death log shows what you couldn't heal through.
- Threat Plates so you notice when your healing pulls a pack.
Per healing class
- Priest: several Priests' Renews can sit on one target now; frames that show heals over time help.
- Shaman: Riptide at 31 points; the default UI has a totem tracker now.
- Paladin: Holy Shock on a 10-second cooldown at 21 points.
- Druid: Wild Growth at 31 points; heal-over-time healing needs frames that show durations.
Healer tier list: healers by role.