One obvious problem I noticed a while back comes up in Romans 12: if you take your own vengeance, you're not making place for God's wrath. Vengeance is the Lord's, and he will repay as appropriate, but if I take that job on myself, not wanting to leave it up to him, then God finds it necessary to leave it up to me. And I'm not able to get it done right.
But in Ezekiel 25 last week, another reason for the folly of vengeance-taking came clear. Edom, Moab, and Philstia all wound up on the sharp end of vengeance because they were vengeful themselves. The measure they measured was measured back to them. They reaped what they sowed. What went around came around.
What I need more than to be avenged is that people and God not necessarily be avenged upon me when I earn it, which I do once in a while. As James wrote, one that judges without mercy will be judged without mercy.
We want to take our own vengeance because we don't think God is around to take care of us, and because we judge ourselves so righteous that we don't deserve for anyone to be avenged upon us. These are both very dumb beliefs, which is why we're always so dumb when we act upon them by taking our own revenge.