Why is the blackmail scene in Fight Club different from the book?

The ending has some reasons for changes. Perhaps some more context may serve to understand other general differences that may make the novel more political and down-pitched).

• in the novel it is unclear if the Narrator is truly free of Tyler. Some scholars have suggested that the last chapter [of the book] may even be narrated by Tyler instead of the Narrator. The implications of such an ending in the film may have been seen by the filmmakers to be too depressing for movie-going audiences.
• Fincher himself has admitted that he wanted the audience to like Tyler but not to miss him when he is gone. Tyler's commitment to his ideology is still strong in the film but Pitt's Tyler is not quite as dark as Palahniuk's.
• When Tyler describes the hunter-gatherer/pre-agrarian society he wishes to return to there is the impression that Tyler vision, while still dangerous, is ultimately intended to free people. In the novel, Tyler gives almost the exact same speech to the Narrator but it is clear that this new world will be led by Tyler.

The purpose of Project Mayhem was also fuzzy in comparison between mediums. Yahoo:

In the novel Project Mayhem was to slow down humanity's technological advancement by artificially causing another Dark Age. This is referred to in the film, however, in the bedroom scene after the car crash.