"Avenge" is usually a verb: "I avenged my brother's death." "Revenge" is usually a noun: "I got my revenge." "Venge" is not commonly said, but "vengeance" is. "Vengeance" is a noun like "revenge," but has a slightly different feel. "Vengeance" sounds a little more poetic/literary to me.

Avenge is simply a verb; while revenge can either be a noun or a verb. Avenge is more of inflicting punishment for a wrongdoing, while revenge can be more of a personal cause and less concerned in justice. It is just focused in assaulting another in return for a similar attack. Lastly, venge is the root word for them. :)