Part One Was Better...
I thought that "The Girl with the Dragon Tattoo" was much better than "The Girl Who Played with Fire."
"Played with Fire" lacks three important elements that were strengthens in the first movie.
1) What I really enjoyed about the first film was the sexual dynamic between Lisbeth and Blomkvist. The partnership, which led to friendship, then love, in the first film wasn't there in the second film. Their relationship could not be developed, because they never share a glance, a kiss, or a face-to-face conversation in the entire film. While the first film felt more or less like a completed work, the second film felt more like a prequel to the third film.
2) The second element that was lacking in "The Girl Who Played with Fire" film was mystery. There was none in the second film. In the first film the audience is trying to determine 'who,' 'what,' 'when,' 'where,' 'how,' and 'why,' like a Agatha Christie novel. Nevertheless, in the second film the only question that the audience has is 'why.' Everything else is presented to the audience in a somewhat straightforward manner.
3) The third element that was lacking in the film was tension. In the first film, I was on the "edge of my seat." I was scared for Lisbeth and Blomkvist. The rape sequence in the first film, got my attention and held it. I was empathetic and scared for Lisbeth throughout the first film. Nevertheless, in the second film Lisbeth is fearless. It's comforting to know that she's invincible in the second film; however, it does significantly diminish the tension.
Cheers!
Helpmann :)