In modern usage, to denounce is to openly condemn (someone) or to publicly proclaim someone to be wicked or criminal. Denunciations are usually quite serious. For example, a president of a country might denounce another country’s leader for unjustly invading a neighbor, or one might denounce one’s boss for corrupt business practices.
To renounce is to give up or reject something, especially formally. For example, a prince might renounce his claim to the throne in order to live a quiet, private life.