At the start of the World of Ruin phase of the game, Celes can feed fish to Cid. Based on the quality of the fish, Cid will either either live or die. How does the rest of the game change based on whether Cid lives or dies?
There's not much difference in terms of the rest of the story - regardless, Celes will leave the island in search of her friends.
If Cid lives, he will send Celes off on the raft he's built, and if he dies, Celes will discover a note that inspires her to take the raft on her own.
After you've left the island, I don't believe the game is different in any way. I think if he lives, you can return to the island later to visit him, but I don't believe this changes the events of the game either.
There's a description of the events on this page:
First - you'll cry so much less if you save Cid. It's really hard to do, it's just luck-based enough that it feels like a real accomplishment when you do save him. The game is not significantly different, but there's a bit of dialogue later that you don't have access to on the small island if you don't save him. It's minor, but I think it's worthwhile.