I have an aeropress, but I rarely use it. Maybe I'm just a clutz, but I always have to wipe the counter down after using an aeropress (especially with the inverted method), while a french press is pretty much mess-free. The aeropress is very fidly. That's great if you're in the mood to experiment but not so great if you just want a decent cup of coffee. I do like the results of the aeropress for making iced coffee, but generally prefer the flavor of french press coffee. Paper filters remove a lot of coffee's flavor-rich oils.

The one deal-breaker that some people have with french presses is the sludge they let through, especially when used with a low quality grinder that doesn't produce grounds of uniform size. If you can't stand sludge and have a crappy grinder you pretty much have to use something with a paper filter.