There are several things from the E3 2011 demo that I'm bummed didn't make the final cut:
1) The puppet shop with the Washington marionettes dangling from the ceiling, and the moment where Liz puts on a Lincoln mask and Booker goes, "Keep looking, Lincoln." Granted, in the final version of the story, Columbia hates Lincoln since he stood up for the blacks and ended slavery, but still, they could've changed the mask to, say, Ben Franklin. At the end of the day, though, I love puppets, and was looking forward to seeing them in the game.
2) The projector showing black-and-white footage against the red banners of the Vox. There was something deliciously eerie about this in the E3 2011 demo, the bits where you'd see Daisy's face projected against the crimson cloth. It was such a creative and iconic sight that I'm surprised they couldn't find a place to use it.
3) The crank-sirens the Vox use to call in zeppelin support. In the E3 2011 demo, they were used after Booker halted the public execution -- another cool bit that was missing from the final game.