Nothing happens really. I played through with and without killing him and it doesn't have an effect on the story at all. He won't send you any goodies like Grace does. Well - you may see different endings depending which other choices you have taken/still will take - check http://www.cheatscodesguides.com/pc-cheats/bioshock-2/

He was responsible for turning Eleanor into a Little Sister and you into a Big Daddy. He messed up the place, he was corrupt.

Would you kill the man who was responsible for what you are now?