Retributive justice is the idea that justice can be achieved through punishment for a crime. So if a person commits murder, they are sent to prison for life or executed. Through the punishment of the individual (which also acts, theoretically, as a deterrent to other would-be criminals) society exacts justice. Modern concepts of retributive justice emphasize that punishments ought to be proportional to the crime. So retributive justice is not achieved by executing someone for theft.
Justice by restitution, or "restorative justice," requires the perpetrator of a crime to make amends for their crime, usually by paying financial damages. Restitution may also include paying medical bills or paying for repairs to property, including automobiles. The idea is that achieving retributive justice is perhaps not enough, that the knowledge that a criminal has been punished for their actions does not provide justice for someone who suffered financial loss as a result of their victimhood.