You'll probably get a more knowledgeable reply than this one, but here's one distinction I myself had to unlearn regarding one difference:
I'd been taught to do a Nat Spin (part 4-6) like a pivot - a thigh-to-thigh rotation (sort of). When I got a qualified Int'l trainer, I properly relearned it as "swinging" the lady past me (with my own LF/hip rotation), then "picking her up" (RF heel lead, instead of sticking my right leg in her way as I had been doing. 
There are other thigns we call "pivots", too - Rev Pivot in I-W or QS, for example, or repeated half-turn pivots down the LOD, or the Promenade Pivot Turn in A-W & A-FT come to mind. These don't use the trained swinging/reaching movements of a spin turn, as I understand (& do) it. Any correction welcomed.