search problem vs optimization problem
The fundamental difference in these two problems lies in the verification of a proposed solution.

The solution of a search problem is only as hard to verify correct as the predicate itself.

The solution of an optimization problem needs special arguments or proofs to avoid having to search the entire domain to verify a solution correct, which often may not exist or are not known.

This means that simply finding a solution is enough for a search problem, but without a proof or exhaustive search an instance for an optimization problem is uncertain.