There is no difference really. Rabbit and bunny are literally the same animal. Some people even say bunny rabbit. Rabbit however is the formal way of referring to the animal. It’s used in texts etc. While bunny is more casual and familiar. It would be more appropriate if it’s a pet or if you’re referring to the rabbit with some kind of affection. 

thank you！

うさぎ、野うさぎ、うさちゃん
いろいろな呼び方があるのと同じような感じなのかなと思いました！