Revenge is a coping mechanism that a person uses to feel better about something bad that has happened to them. It consists of believing the lie that someone deserves to be punished for doing something bad, and that this is just.
Revenge is never justice. The fallacy here is the irrational belief that the person who did something wrong deserves punishment, but I do not believe that there is any basis for this. Doing a second bad thing to satisfy your own urges can in no way justify something that happened to you. The idea that the person deserves to be hurt, or that the universe, life, or anything else is out of balance until they suffer is entirely a fabrication resulting from trying to find a reason for what happened to you, irrational misinformation, and anger. Anger in this situation results from the need to blame something other than yourself to cope with the situation, and is therefore not reasonable, but an emotional reaction to lessen pain. Since this is just an instinctive reaction, it can’t be trusted to be reasonable or true.
Justice is by definition what is fair. Punishing someone is never fair, it’s another bad thing that people tell themelves is justified. For a bad thing to be justified, it has to be outweighed by a good thing. The only good thing that inherently comes from revenge, sometimes, is a feeling of accomplishment and restoration of balance, however the goal that was accomplished and the perceived imbalance are both fictional, created to reduce pain. Giving someone a goal, such as revenge, is a way for them to take their minds off of the pain they have suffered and also gives them an objective that promises to fix things, atleast to a certain extent.
If someone can create the illusion of vengeance being necessary in order to cope with pain, then they are lying to themselves which is unhealthy and will most likely cause more problems. If they were to ask themselves why someone deserves to suffer from vengeance, I do not think that they could find a logical answer. In the end, a person who takes revenge has lied to themselves to avoid dealing with pain and desperately looking for a way to fix what has happened to them, and either realises that their revenge was unjust and pointless, or lies to themselves further and thinks that it was justified.