Is there a difference? Apparently slightly so!
According to Merriam-Webster, in this context:
A rap is "a negative and often undeserved reputation or charge —often used with bum or bad"
A rep is slang for "reputation; especially : status in a group (as a gang)"
So, from these definitions and other things I've read:
- If someone has a "bad rap", it's typically because that person is getting blamed for or associated with negativity for something that person may not have done.
- If someone has a "bad rep", others also think poorly of that person, but likely for something that person did, or for a track record of sketchy behavior.