It CAN make a difference, depending on each show's unique circumstances.
For WICKED, it didn't matter that it lost.
But for AVENUE Q, I think winning the Tony helped make it a "must-see" phenomenon. Q opened earlier in the season and, if I remember correctly, was slowing its box office draw until it got Tony attention.
"The Spectacle has, indeed, an emotional attraction of its own, but, of all the parts, it is the least artistic, and connected least with the art of poetry. For the power of Tragedy, we may be sure, is felt even apart from representation and actors. Besides, the production of spectacular effects depends more on the art of the stage machinist than on that of the poet."
--Aristotle