They are close but not the same. An enlisted guy can have more than one NEC but an officer will have only one designator.Enlisted have designators, too (NEC codes), um...sir .
Designator = warfare specialty. For example, 1320 = NFO. P-3, F-18, EA-6B, etc. NFOs all have the same 1320 designator.
NEC = specific qualification within their specialty (rate). For example, AWs. A P-3 SS1 has a different NEC than a P-3 SS3. But an AW can hold both NECs.