Oh, this happens to me a lot.
Occasionally it is just a product of viewing size–the gap appears when viewed at some scales but not others, and won’t appear in the final image.
However it may be an anti-aliasing issue–when photoshop turns your path into a selection (Which it does to accomplish the layer via cut) it makes two nice soft edges which might not add up to full opacity at the seam where they overlap.
Instead of doing layer via cut, you can duplicate the WHOLE layer, and use different layer masks on the copies to isolate different regions–this way you can fill the gaps by finessing the layer masks.
Or just do the masking in After Effects–not sure if that will slow down your animating, though.