When revenge is a verb, there is no difference between it and avenge. Both mean to seek or get vengeance. Yet while this use of revenge is many centuries old, the word is now rarely used this way. It tends to give way to avenge in the verb use.
While revenge can work in place of avenge (but rarely does), the inverse is not true. Avenge does not work as a noun, so it doesn’t work in place of revenge in the latter’s usual use.