I've always thought of 'avenge' as the verb form of revenge.. somewhat similar to effect/affect. By avenging something, you get revenge. In fact they're almost equivalent. Spiderman swore to avenge his uncle's death. = He swore he'd get revenge for his uncle's death. However, I think that one way they differ is that avenge has a more serious tone. Revenge can be used more lightly, such as if you say to a friend "I'm going to get revenge, just watch out!" in response to a prank he pulled, let's say. You could say that as a joke, without really intending to get revenge. Maybe it's because the word has been overused, I don't know... but that's my experience with using the word haha.

Avenge is a verb by the way so that should make it way easier to understand hahaha. Where as revenge is the act of getting someone back for an injustice done to them. Example, Avenge me! I will get revenge!

These are so very similar. I'd suggest that "avenge" has a sense of justice; "revenge" does not, and is therefore more harsh and driven by hate.