If you use LEM to prove A, you can translate that proof into intuitionistic logic so that it proves Not (Not A). This is classically equivalent to A, but not in

If you use LEM to prove A, you can translate that proof into intuitionistic logic so that it proves Not (Not A). This is classically equivalent to A, but not intuitionistically. So the real difference between the two is that intuitionistic logic makes a finer distinction between propositions that classical logic says are the same.