All vertebrates possess three salient features of chordates:
Presence of Notochord.
Presence of dorsal, hollow nerve cord.
Presence of pharyngeal gill slits in some stage of their life cycle. Hence all vertebrates are chordates.
All chordates are not vertebrates:
Primitive chordates have only a notochord which is not replaced by vertebral column as seen in vertebrates. Hence all chordates are not vertebrates