A. Revenge is a subjective, personal, wild form of justice, untempered by objectivity and truth. Often the focus of revenge is a sense of satisfaction or enjoyment, while punishment is about consequence, and a deterrent and hopefully also an attempt to rehabilitate. Revenge is more about our emotional response than about what needs to be done to correct the situation.
Most punishment is administered by society or those entrusted by society or law (i.e. parents, teachers, police etc..) while revenge is more personal. By the way, the Torah condemns revenge but consequence, reward and punishment are very much part of our books. (Although Chassidus teaching, vs. Mussar, puts much more emphasis on positive encouragment and inspiration than negative punishment).