To vanquish is to be the complete and total winner, to overpower and overcome, whether in a contest, a race, or a war. It generally suggests a total trouncing, to the point of humiliation — or worse — for the loser.
Sometimes words for the same thing are effective in different ways because they offer different levels of meaning. For example, in a game, you can simply win, or you can vanquish your opponent. The former is enough, but the latter makes that defeat sound so much worse, like a total rout. Philosopher Friedrich Nietzsche once stated, "Against war it may be said that it makes the victor stupid and the vanquished revengeful."