The verbs avenge and revenge mean basically the same thing, but avenge is widely perceived as having nobler connotations than revenge.
Both words, like the English word vindicate, derive from the Latin verb vindicare/vendicare: “to claim, to set free, to punish.”
In early usage, all three words, vindicate, avenge, and revenge meant, “to punish” or “to exact retribution.”
In modern usage, the most common meaning of vindicate is “to clear from censure, criticism, suspicion, or doubt” or “to justify.” People and things may be vindicated:
Mubarak’s Wife Says Husband Has Been Vindicated
Bayliss insists the decision made before his appointment to retain England’s leading Test run-scorer as captain has been vindicated.
A year after his shock resignation, Pope Emeritus Benedict has no regrets and believes history will vindicate his tumultuous and much-criticized papacy.
Avenge and revenge, however, retain the ideas of punishment and retaliation.
Note: The word revenge functions as both noun and verb. Avenge is always a verb. Its noun form is vengeance.
Although there is no authoritative rule to distinguish avenge and vengeance from revenge, a perception exists that there is a difference. For example, The Chicago Manual of Style offers this note:
Avenge connotes an exaction for a wrong. The corresponding noun is vengeance. Revenge connotes the infliction of harm on another out of anger or resentment. Revenge is much more commonly a noun.
Likewise, Paul Brians (Common Errors in English Usage), promotes the distinction:
When you try to get vengeance for people who’ve been wronged, you want to avenge them. You can also avenge a wrong itself: “He avenged the murder by taking vengeance on the killer.” Substituting “revenge” for “avenge” in such contexts is very common, but frowned on by some people. They feel that if you seek revenge in the pursuit of justice you want to avenge wrongs, not revenge them.
Two examples from literature illustrate the difference. In The Princess Bride, Inigo Montoya seeks vengeance for the murder of his father, whereas in Moby Dick, Captain Ahab seeks revenge for an injury inflicted by an animal. Montoya has justice on his side, but Ahab acts from wounded pride, irrationally attributing vindictive intentions to a nonhuman creature.
Considering how mean-spirited cyber culture has become, the distinction between revenge and avenge is probably a useful one to observe. Here, for example, are the titles of just a few of the distressingly large number of how-to articles on the topic of punishing people for perceived affronts:
How to Get Revenge on Anyone
The Ten Commandments of Revenge
50 Random Ways To Get Revenge
How to take revenge on your Boss
10 Outrageous Ways to Get Revenge On An Ex
How to Get Subtle Revenge on Your Enemies
How to take revenge on Your Parents
How to Get the Sweetest Revenge Ever
Note: Until recently, the phrase “take revenge” was more common than “get revenge.” “Take revenge” is still more common than “get revenge” in the Ngram database of printed books, but a Google search shows “get revenge” ahead of “take revenge” on the Web.
Perhaps we should reserve vengeance and avenge for retribution motivated by a wrong that any reasonable person would regard as appalling and use revenge to denote the desire to hurt someone for no better motive than anger or hurt pride.