With a French Press you're using a course grind with a metal screen, and hotter water that needs to steep longer (about 4 minutes). The AeroPress uses fine grind (espresso grind) and the ideal water temp is about 170-190, and you only steep it for 30 seconds. The paper filter takes out the grinds better. If you have a hot water dispenser on your home sink or office water cooler, that's an ideal setup for near-instant brewing with the AeroPress.

So the big difference is cooler water, shorter steep time, finer grind, and better filtering. = less bitter, fuller richer taste, less sediment.

There are 3rd parties selling stainless AeroPress filters but I haven't tried one. The AeroPress paper filters are very thin and small, organic, inexpensive and easy to pop out into the garbage with the used coffee puck. (I think there is more upside to the paper filters than a metal one which would need separate cleaning and delicate handling to separate from the hot used grounds.) You get about 300 filters for a few bucks, so cost isn't a factor.