They are very similar, but I would say that "triumph" is a greater, more spectacular victory. "Prevail" refers to simply doing better than the opponent, more like "survive" or "succeed".

There's a lot of overlap. However, we usually "triumph" in contests where the reward for winning is more important than the penalty for losing: "It was a tough game, but in the end our team triumphed!" We usually use "prevail" where there are larger penalties for losing, especially if losing means that you cease to exist. "In the past, 5 companies made this product, but only one company prevailed. It bought or out-competed the other companies,and now it has a monopoly."