Generally speaking, each side offers a different mode of relaxation. The more you like to socialize and party on vacation, the more the Dutch side plays in your favor. Maho Beach is where you'll find tourists throwing back cold ones during the day, and, along with Simpson Bay, it's the major source of nightlife on the island in terms of nightclubs (Tantric and Bliss), casinos (there are several in Maho and Simpson Bay), and late-night hangouts (most of which are concentrated on the main drag of Maho). You can get a drink on the French side, of course, but it's much more mellow, such as at the kick-back beach bars of Grand Case, shown above.
Since Maho is one of the most popular places thanks to its nightlife, there's something else that should be said: It really isn't quiet during the day, either. But not because of partying -- it's because of the airport. From morning until after dark, you will hear the airplanes loud and clear, and, at times, you'll even smell the exhaust fumes. It's not the end of the world, so long as you are prepared for it and know it going in. This past visit, we met several people who were "surprised" at how much noise came from the planes, saying that it prevented them from relaxing. Another thing to keep in mind is that Philipsburg, the capital of the Dutch side, is where the cruise ships port, so those staying there will be joined by passengers from two to five ships on any given day.
So, in summary: Dutch side = party; French side = tranquil. Pick your poison!
[Photos: Will McGough]