To avenge is to exact retribution for a wrong, usually on behalf of someone else and with a sense of justice — avenge his brother's death. Revenge is retaliation taken for personal satisfaction. Avenge is chiefly a verb; revenge is chiefly a noun.
The core difference
Both answer a wrong. One claims justice, the other claims satisfaction.
- avenge (verb) — exact just retribution, often for another: she avenged the insult to her family.
- revenge (noun) — retaliation for personal satisfaction: took revenge.
The grammar
Avenge works as a verb and gives the noun vengeance. Revenge works mainly as a noun, though it exists as a verb in the reflexive — revenge oneself on. So "he revenged his father" is awkward; "he avenged his father" is the idiom.
What the distinction carries
The words differ in whose interest is served. Avenging is presented as righting a wrong; revenge is presented as satisfying the wronged party. That is why literature reaches for one or the other deliberately.