Quitting Emmesmail tells the program that your session has gone well and Emmesmail should now finish up. This means Emmesmail should save all requested files, delete all requested files, and modify the users mailbox accordingly. It also should send out any emails queued to be mailed.
Exiting, on the other hand, tells the program that the session has not gone well, or you have changed your mind about what you want to do. Exiting attempts to leave the system in the same condition it was in at the start of your session. It does not delete any files, none are saved, and nothing is mailed, regardless of what was requested during the session.