Hello Fansico "vengeance" come from latin "vindicare"

They are not completely synonymous. "Revenge" is more spiteful, more punitive, injurous. "Vengeance" is more retribution for the sake of restoring justice. Like: "The vengeance of the Lord was upon him." You couldn't use 'revenge' there, really. But you can say: "When his family was slaughtered, Charles Bronson took revenge in his usual style, killing everything and everyone around him."

They are synonyms. 'Take vengeance on / upon somebody' (formal) and 'take revenge on somebody' mean 'the act of punishing or harming somebody in return for what they have done to you, your family or friends. It is something that you do to make somebody suffer because they have made you suffer'.