A variable ND filter works on the same basic principle as two polarizing elements rotated relative to each other: as their alignment changes, they pass less light.

So in theory, a variable ND is essentially a controlled pair of polarizers built into one filter. The practical differences are mostly in implementation and side effects rather than a different optical principle.

One issue noted with variable polarizers is with very wide-angle lenses: you can get uneven darkening or banding because light hits the filter at very oblique angles across the frame. If you stack two separate polarizers, the greater spacing between them can make this effect worse.

So the key difference is not that a variable ND uses a fundamentally different theory, but that a dedicated variable ND packages the polarizing elements more tightly and predictably. That can help reduce some problems, though wide-angle banding can still occur.