Biopunk is a sub-genre of cyberpunk, but more to do with, as others have said, DNA modification and such. Cyberpunk without the biopunk has more to do with hackers and themes like digitizing human consciousness, and body modifications on a purely physical level (androids as opposed to mutants). I associate a style of graphic imagery with it.
An example of biopunk would be the Hojo-related parts in Final Fantasy VII. In Hojo's lab, there are many genetically-altered creatures that have a fleshy, mutilated-body sort of look. Hojo himself becomes an example of biopunk near the end of the game.