You can prove more stuff with classical logic while intuitionistic logic restricts you.
For example given a real number x constructed in intuitionistic logic. You can't determine if x > 0 or x = 0 or x < 0. While you can in classical logic.
Also, more generally you can't prove existence statements in general without construction in intuitionistic logic.
So, there exists an x such that P(x) can be proven without actually finding x classically, but in intuitionistic logic I must provide a procedure for constructing x.
All this said, even though you can prove less statements in intuitionistic logic I find it's restrictions satisfying because it forces us to prove things by showing they exist via construction. Which to me is more satisfying than just showing that a construction exists.
You can prove more stuff with classical logic while intuitionistic logic restricts you.
For example given a real number x constructed in intuitionistic logic. You can't determine if x > 0 or x = 0 or x < 0. While you can in classical logic.
Also, more generally you can't prove existence statements in general without construction in intuitionistic logic.
So, there exists an x such that P(x) can be proven without actually finding x classically, but in intuitionistic logic I must provide a procedure for constructing x.
All this said, even though you can prove less statements in intuitionistic logic I find it's restrictions satisfying because it forces us to prove things by showing they exist via construction. Which to me is more satisfying than just showing that a construction exists.