They are equivalent in zero-sum games, but not in non-zero-sum games. For example, consider a game between Alice and Bob who are sitting on a bomb. Each of them has three options:

1) Enjoy a nice latte

2) Trigger the bomb, killing both

3) Disarm the bomb, stopping the other from triggering it

The min-max strategy is to disarm the bomb, but there's a better Nash equilibrium where both players enjoy their lattes.