I spend a considerable amount of time adding removing iMessage from my iPhone and iPad so that it would have the same messages on both devices. I'm still not sure what obscure combination worked.
However, if you want to keep them separate it is simple enough. Go into iMessage settings on the iPhone, tap the Receive At option then tap the Apple ID and sign out. This should remove the Apple ID email address, but leave the phone number.
I'm almost certain this will work. I just finished fiddling with FaceTime and that is how it works. However, if it doesn't, just sign in with the Apple ID again to be back where you started, which is working at least; if not the way you want.
Edit: this will stop the iPhone from receiving both by phone number and email (as Leelai described).