The dictionary defines rabbit as:
"a burrowing, gregarious, plant-eating mammal with long ears, long hind legs, and a short tail."
The dictionary defines "bunny" as:
"a rabbit, especially a young one."
"Bunny" and "Rabbit" are the same thing. Especially in the "bunny" definition just means that it's commonly known that way. If a bunny would be classified as an adult rabbit or a young rabbit, it would be the young rabbit.
"Bunny" is basically slang for rabbit. "Coney" also means rabbit - however, it's an old-fashioned term, yet still commonly used in England.
So no, Rabbit and Bunny are not different. They are one and the same.