What is the difference between vanquish and overcome?

You might get a better or more accurate answer in a words & wordplay post, but to my understanding, vanquish implies de-activating the other person or force - pulling the plug on it, rendering it "off" or finished. Overcome means temporarily getting the upper hand, but not necessarily putting the other person/thing's lights out.

They mean about the same thing. To me, vanquish implies using more force.

One overcomes a personal problem and one vanquishes an enemy.