An NP-hard problem differs from an NP-complete one because NP-hard problems

NP-hard requires only that EVERY NP problem reduces to it. The hard problem itself may sit outside NP entirely. The Halting Problem is NP-hard but not even decidable, so certainly not in NP. **NP-complete adds the second requirement:** the problem must itself be in NP. Polynomial verification, P-membership and coNP-membership are not part of NP-hardness.