I would say Justice is not about punishment, but more about rehabilitation, and that's what I would call the difference.
If we can all agree on specific definitions, you're partly right and partly wrong. Justice has little to do with rehabilitation.
Justice -noun
1. the quality of being just; righteousness, equitableness, or moral rightness: to uphold the justice of a cause.
2. rightfulness or lawfulness, as of a claim or title; justness of ground or reason: to complain with justice.
3. the moral principle determining just conduct.
4. conformity to this principle, as manifested in conduct; just conduct, dealing, or treatment.
5. the administering of deserved punishment or reward.
6. the maintenance or administration of what is just by law, as by judicial or other proceedings: a court of justice.
Revenge -verb
1. to exact punishment or expiation for a wrong on behalf of, esp. in a resentful or vindictive spirit: He revenged his murdered brother.
2. to take vengeance for; inflict punishment for; avenge: He revenged his brother's murder.
3. to take revenge.
–noun
4. the act of revenging; retaliation for injuries or wrongs; vengeance.
5. something done in vengeance.
6. the desire to revenge; vindictiveness.
7. an opportunity to retaliate or gain satisfaction.