Does it make any difference if I choose the Bird or the Cage Brooch?

Well, not directly, so don't worry about it.

The point here is that you made a choice that you will remember and notice it later on.

More information with SPOILERS, read only AFTER FINISHING THE GAME:

Thanks to this choice you will almost immediately notice that the girl at the end is not Elizabeth since she doesn't have the brooch you selected for her. It was a nice touch by the game creators.

Needless to say – SPOILERS AHEAD!

Like many other choices in the game, it has no effect on the gameplay whatsoever. However, the act of choosing a brooch has a significant bearing on the story.

In one of the final scenes, Booker will recognize that "Elizabeth" is not the Elizabeth he's been trying to rescue. A giveaway is that this Elizabeth will not be wearing the brooch that Booker chose for her.