The flip side is that you lost the ability to "program" a solution. And they are not always simpler. Good constructive proofs usually goes by "Look at this object, now back at me, now at this object, I'm on a boat, I do a little computation, I'm on a horse, I do a little reduction, and I end up with the result I want, Qed."

That is, they start by claiming that a given object has the property you want and then proceed to show that it is the case.

I'm not a constructivist by any stretch of the word, but knowing different kinds of proofs of the same object is often giving us insight which we wouldn't have otherwise.