My initial thought was that it was a problem with the wood filler, perhaps being the wrong color. However, I later realized the real source of the problem. The wood filler was fine, in and of itself. However, what was happening was that the wood filler was seeping into the pores in the wood surrounding the nail hole, and even though I was later sanding the filler flush with the wood, the filler that seeped into the stock around the hole would not stain the same color as the stock, and it stood out like a sore thumb on the finished piece.
With that in mind, I realized I had to learn how to deal with the problem. I determined that because wood fillers contain moisture, they're going to work themselves into the surrounding wood if left on for any period of time. I could try and remove any excess immediately, but because wood filler shrinks (slightly) when it dries, I would likely need to re-apply the filler to cover up the shrinkage from drying.
I have since learned to apply the filler before beginning any sanding on the project whatsoever. When I sand the project (using progressively higher grits of sandpaper), I will end up sanding off not only the excess filler, but also a very thin layer off of the top of the surface of the stock, which is usually deep enough to remove any filler that has seeped into the pores of the stock. While the filler in the nail hole still stains a slightly different color, it is much more difficult to spot since it is only the area of the nail hole instead of about a half-inch diameter area that was initially covered by wood filler. The result is a much more evenly stained finish.