The rabbits they refer to are different. The word 'Bunny' is generally used to describe cute little rabbits, which is a nickname, while the word 'rabbit' generally refers to all rabbits, especially those with small size and short tail and pet rabbits.
Many people with rich English vocabulary or who like watching English dramas and American dramas should know that there are many words used to call rabbits in English, but not all of them can be used to call all rabbits.
For example, the word 'Bunny' is generally used to describe the cute little rabbit. It can be said to be a nickname, just like the word for dog in English has' puppy 'in addition to' dog '.
The word 'rabbit' refers to all rabbits, especially those with small body and short tail. There is also an English word 'Hare', which is also used to call rabbits, but this word specifically refers to rabbits.