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I've not read any Larsson yet, but I have read Lagercrantz's "Spider's Web" and thoroughly enjoyed it.
I've seen the "Dragon Tattoo" film of course.
Reviewers seem to agree that the second author is the better writer. I like his compact volumes and have been rather put off by Larsson's biblical size-and-weight tomes.
I also understand that Larsson wrote dark descriptions of violence which Lagercrantz avoids. 
Good! I have never been fond of violence.