Nothing, technically. But
Amy could be used as a nn for
Amelia or
Amethyst.
In fact, a long time ago I read a short children's novel where a queen named all her daughters after jewels (
Ruby,
Pearl,
Diamond, etc); the youngest, the heroine, was named
Amethyst.
Princess Amethyst called herself
Amy as a nn, because she wanted to be different from her older sisters.
Miranda