I believe a protein skimmer removes, through the bubbling process, the organic material(s) from Urea and uneaten foods that make up ammonia before it breaks down and becomes nitrites then nitrates. (It works at a different level than your biological filter.) So the skimmer doesn't actually break anything down, it removes it from the water. This is why you have the lower nitrate reading. Without ammonia, you can't have eventual nitrates in the tank. If you get rid of the ammonia before it breaks down, you basically have no nitrate source.
Considering that most skimmers will not get all of the product out of the water, this is where your biological filter comes into play to actually break down the "missed" ammonia and turn it into a more harmless material. The biological filter is what breaks down fish waste, not the skimmer. This is why both filtering types are highly recommended in an aquarium.
Hope this better explains it