Post-hardcore are bands like Alesana, metalcore can range anywhere from thrash oriented angst like Bullet for My Valentine, to melodic death metal with a dulled edge like Parkway Drive or full on pop music like Asking Alexandria, Attack Attack and other crabcore bands.

Metalcore is "supposed" to have metal elements, post-hardcore is what it exactly says on the tin. Either literally post hardcore or post-rock + hardcore.