Is there a difference between justice and revenge?

Justice, in (my interpretation of) the strictly legal sense, is the achievement of the wishes of the population with regard to a person or group of people’s actions, the consistency of which is maintained by the application of pre-dating guidelines. Since it is the wishes of the population being carried out, it is the vengeance of the population on the criminal. Is there any difference between justice and revenge? Where does our ‘sense of justice’, if we have one, come from?