He has either the best or worst secret identity management skills in the business.This has lead to some of his friends suffering, but, to be fair, this also happens a lot to his more secretive colleagues. In a world full of cameras and databases, there's much to be said for the open, humorous approach.
I mean, can you imagine Clark Kent doing that? Granted, he has much more powerful enemies, and it wouldn't be compatible with either Superman or Clark's public images as they currently stand — you can't really be Superman (I thought about using words like "the world's most beloved superhero", "iconic", etc, but, no, the right phrase is "you can't be Superman", which is both all of this and more) if people know you have a day job and lived in Smallville, USA. Hell, the US would reinstate mandatory military service just to get to give him orders.
So, really, there are practical issues to this idea. Bruce could be an open Batman more easily. He breaks the law every night, but have you seen his lawyer pool? If Luthor gets away with trying to take over the world every other week, Wayne would definitely get away with saving people all the time. Plus, most of his big enemies already know who he is, so that cat (pun not intended) has left the bag.
But Bruce, professional trinity-level badass that he is, doesn't have the kind of godlike power Clark does. Nobody particularly wants to use him, either as weapon or for propaganda; if anything, they'd like him to go the hell away.