Anya knows from the beginning that Dmitri is out for a reward for bringing her in, though he downplays it. So at the ballet when they had the big blow-out fight in the movie, she's instead annoyed at herself for going along with Dmitry instead of being furious at Dmitry.
The fact that she knows who he is all along also cuts a lot of the tension between her and Dmitry, since he's not trying to hide who he really is.
Another source of tension that's cut is Dmitry knowing that Anya is the princess while Anya herself doesn't know - in the show they realize it simultaneously. (That moment is my absolute favorite of the show and has fantastic emotional chemistry, but it's a short moment.)
Finally, at the end of the movie, gives up Anya because he doesn't fit in her new life, gets in line for the train, then pulls out a rose and decides to come back for her (just in time to help fight off Rasputin). In the show, he doesn't come back - Anya comes after him.