Flexibility- You can make Espresso style and Filter Coffee
Easy- the Aeropress is incredibly easy to use and very forgiving- French Press will punish you with horrible taste if you make a slight error. It's very hard to get consistent result with FP (imo
)
Ease of clean up- the ease and speed of clean up in unrivaled. And this does become an issue once the novelty of a new brewer wears off and you're making your hundredth cup etc.
Aeropress with the Hasbean filter method give you a very strong cup of very tasty Coffee. You can then dilute it to your taste. The taste is different to that of FP. The paper filters of the Aeropress can impart a somewhat papery taste (to some people, can't say that it has been an issue for me). It does give you a clean cup of Coffee, with no grounds or fines (powder from griding Coffee) finding it's way into your cup. Coffee oils (tasty) are blocked by paper, wheras they would not be blocked by a metal filter. Paper does hold back those oils, which can increase your Cholesterol apparently, but it does leave you with a lovely clean cup.
You can make your Coffee faster in an Aeropress, but not a great deal faster, so it's hardly a dealbreaker.