In the example you've given, I can't think of any discernable difference.
However, in other contexts, the words are not interchangeable. When dealing with motion, use stop:
You need to stop the car! (not: You need to quit the car!)
When dealing with habit-breaking, stop is acceptable, but quit is often the better word:
You need to quit biting your fingernails.
When leaving a place of employment, you normally quit the job, but you might stop working someplace. In other words:
How are things going at the restaurant?
Oh, I quit that job.
or:
How are things going at the restaurant?
Oh, I stopped working there.
Such differences in meaning are too numerous to exhaustively list here, but you can learn more by examining a dictionary. Collins is pretty good – see: quit; stop.