The only difference is that a red paint will "seem to cover" a gray primer somewhat more easily than a white primer. That is, it will take fewer coats to make it look good.
All paints allow a certain degree of "print-through". That is, if you painted a checkerboard with 1 coat, you could still see the checkerboard pattern. That is just a practicality of how paints are made. The answer is more coats - with enough coats, even the checkerboard would be hidden.
In a perfect world, you make the primer exactly the color of the topcoat. It will print through just as badly as the checkerboard. But you won't notice! LOL!
So that's what the gray is all about. Color has 3 traits:
- Chroma (saturation)
Lightness tends to be what prints through strongest, so the goal with gray primer is to get the lightness closer. So that print-through will be less obvious.
For instance, I have both white and gray primer in one paint system. When painting yellow, I mix the primers about 6:1 (a little gray goes a long way) giving a light gray, probably a Munsell 9/ in lightness. (0-10). It works better with the yellow than white does.
Red is particularly difficult, since good pigments are Just Not Available anymore. You probably don't want cinnabar, since it's mercury. So reds tend to be worse than average with print-through.
So, what does "more difficult" mean? It simply means more coats to get good coverage.
Overcoating will work, provided you follow the manufacturers' instructions regarding compatibility and cure times.