Revenge is emotional. Justice is logical.
An individual may wish to kill someone in the throes of emotional distress and cry "Revenge!", but lacks the authority inherent in government to execute such a sentence. This way lies Committees of Vigilance and degenerates into lynch mobs.
Justice is the dispassionate execution of a lawfully applied sentence as determined by a jury of the accused peers, within written and argued law as it stands at the moment of trial.
Revenge is almost always individual.
Justice is almost always societal.
Ignoring arguments about whether or not juries and judges are cold-blooded, what does it matter provided that the revenge is not misguided? The results are the same.