There are plenty of times in math classes where I've been assured that "Theorem 1 guarantees that there exists a unique solution...", but where this solution is in no sense easily known.
As an example, a maximizer of some convex function over a convex set. Sure, it exists, but constructing it is another matter!
As another example, solutions to differential equations of a given form. The form guarantees a solution exists, but the solution may difficult to pin down.
One more example: Borel sets.
Hmm, now that I'm going: the square root of 2. It is a number that satisfies a simple property, but constructing it is considerably more difficult.
Not so sure about that. I'd also dispute that they are inherently more difficult (assume X doesn't exist... )