I have to say I was very pleasantly surprised by this version. I liked the original quite a bit, and feared a remake, thinking there was no reason to do it. Yet this film is a better made movie in every way, including the casting of almost every part, but the American version does have one fatal flaw; the explicit violence of the original is much more affecting.
The narrative in this film is much better, as is the visual design (although typically for Fincher, I had problems with both the light and the dialogue levels being too low). However the sound design was outstanding, and the music was surprisingly affective as well.
My major hope when I heard they were doing American versions was that they would make the sequels better than the comparatively weak originals. (Full disclosure, I have not read any of the books.) I still have that hope, and now it's even higher. Unfortunately since this film did not do well in the US, there's a chance there may be no sequels.