Revenge is the word we commonly use when we mean to say that we are doing something wrong because wrong has been done to us, an act of returning evil for evil done to ourselves.
He is raging in anger and wants to take revenge from his colleague who ruined his presentation and took away his chances of getting the promotion.

Avenge is the word which is less commonly used and means that a person is doing something wrong to balance out the wrong done to others, taking revenge for the injustice done to others.
Hamlet wanted to avenge his father’s death which was a sheer act of disloyalty.