Here is a rundown of the differences in the two movies:
-MIKAEL BLOMKVIST-
American - Blomkvist is played as more of a tough guy and not a good guy. His flaws are laid bare and he shows himself to be much more detached than emotional.
Swedish - This is the "good guy" side of Mikael. He is sensitive, caring, and smart. He shows a protective side when it comes to Lisbeth. Physically speaking the Swedish Blomkvist doesn't look as sturdy as his American counterpart. He has a gut and appears to be quite a bit older than Lisbeth which can make the relationship between them more shudder inducing and probably accounts for why there are fewer sex scenes between them in the Swedish version.
-LISBETH SALANDER-
American - Perhaps because Blomkvist was made into such a strong character Lisbeth was then morphed into a more withdrawn and vulnerable girl so as to complement the new Blomkvist. She still has attitude, aggression, and rage but she also exhibits a quiet shy side that was not in the original as well as more of a romantic side.
Swedish - In this version Lisbeth is not shy, not gentle, and not nice. She doesn't chase Blomkvist - he chases her. She perfectly embodies everything you think of when you think of a strong female lead and has an unpredictability and edge to her that is exciting to watch. Her dragon tattoo is much, much better.
-OVERALL-
I liked the American Mikael and the Swedish Lisbeth.
While I may prefer a scene or two from the Swedish version, such as the ending, overall I enjoyed the American version more.
On the flip side, I can understand why some may hate this version because Lisbeth was their favorite character and she's been changed into something they don't like. For me, the modifications to Lisbeth's character weren't severe enough to put me off.
The Swedish version captured a cult following for a reason and I would recommend both to anyone who has an interest in darker gritty movies that have a raw intensity to them. The Girl with the Dragon Tattoo isn't for the faint of heart and that's what I love about it!