Virtually all the positive word of mouth for Curious Incident came from its London run. Yes, it was a best seller in the states, but it was an unconventional book and it might have been slow for audiences to warm to had it not been for the publicity it received through the Olivier awards.
The Olivier awards simply provide another way to market a Broadway Show. For a show like Groundhog Day, which is struggling to set itself apart for other shows in a crowded season, it CAN be important. For what it's worth, I get regular reports from friends in New York on some of the new shows (to help me plan my one trip to New York every year), and although they rarely agree on everything, the consensus is that they loved Groundhog Day, hated War Paint, and had mixed feelings about Amelie. Now, no one is going to put "Back Row's Friends Loved Groundhog Day!" on the marquee because no one cares, but "Olivier Award Winner - Best Musical" is as good as a pull quote from Brantley when it comes to selling tickets. It may or may not be enough to propel the show to commercial success on Broadway, but it is far from meaningless.