Well, I have very little (good) experience with the French press, so I won't even pretend to be able to make a comparison there, however I am quite impressed with the Aeropress, so here are a few thoughts on its use. (I guess I will have to update the sticky on this one of these days).
The way you use the Aeropress makes a very big difference in the results you get, in spite of the fact that it looks so simple. There may well not be a "right way" to use it, it will depend upon your taste and preferences.
Firstly, the grind makes a very noticeable difference, maybe not as much as with espresso, but certainly significant. If you grind too coarse the water just flows through without resistance and the brew is often rather watery. If you grind too fine it tends to plug the paper filter and the brew is often somewhat bitter. If you're using a gold filter grinding too fine will allow silting in the cup. Most people seem to prefer a grind just a bit coarser than for espresso, a very general guideline, to be certain.
Secondly, the water temperature makes a very large difference. Boiling water makes a terrible brew (imo) and water less than about 70°C gives a weak cup, but may be interesting for certain coffees.
Thirdly, pressing immediately after adding the water seems to be preferable to steeping as most people do with the French press.
Fourthly, if you want a longo, don't just run more water through the press, add the pressed coffee to a cup of hot water.
Obviously, parameters like amount of coffee and volume of water make a big difference, but that should be obvious to most of us.
I think that the Aeropress gives one an unusual opportunity to investigate the really subtle flavours in coffee and changing the variables above may provide some really interesting new aspects to coffee enjoyment. As I said above, I don't know if this is any better or different than with the French press.