From what I've gathered, the higher a spirit's rank, the greater it's starting stats.
For example, I have two Komory Bats in my current file. Both are Riku tutorial Komory Bats that I've hardly used (This is a New Game + file) and one is a rank E while the other a rank C.
Komory Bat Rank E:
Attack 7
Magic 10
Defense 5
Komory Bat Rank C:
Attack 12
Magic 15
Defense 9
As you can see the Rank C Komory Bat's Attack and Magic are 5 points higher than the Rank E's and it's Defense 4 points higher.
I've also read (in posts elsewhere by other players) that higher ranked spirits are more focused and perform more actively in battle and their link gauges fill faster.
I currently have two Stars and an A in my parties and although the battles were going to fast for me to focus on it, I could swear that at a couple of points when my HP was critically low I'd suddenly be healed before successfully shuffling curaga to the top of my command deck and using it. I could have been misperceiving and perhaps just caught some HP prizes but I really don't feel it was the latter and if I am right then one of my spirits had to have healed me since there is no Donald in this game. If I had to guess I'd say it was my Aura Lion.
To my knowledge, abilities and ability boards are not altered. The latter can be altered but by disposition, not rank.