Is Bruce Wayne batman? Or is batman Bruce Wayne?

Bruce Wayne is Batman

Batman plays Bruce Wayne, to fascillitate his work as a vigilante.

It was a big difference between The Dark Knight, and original Batman-lore, Christian Bale played Bruce Wayne as someone who struggled to balance the 2 Worlds(which is actually much more Oliver Queen/Green Arrow).

Original Batman is perfectly organised, he's a smooth, slick Billionaire by day, so he can walk freely at night(Michael Keaton did a great version of that in live-action).

Bruce Wayne used to be just Bruce Wayne. However, after the trauma of his parents being murdered, he became Batman. Bruce Wayne, the flamboyant billionaire is just his cover by Day. In reality, Bruce is introverted and he becomes Batman at night to fight the injustices he sees in Gotham.

I like Nick's answer. To expand on that point, there is a particular scene which is telling (if you go by the film's interpretation, which I do in this case) in the film "Batman Returns" wherein Bruce Wayne and Selena Kyle are attendees at a masquerade dance, where upon everybody else at the party is wearing masks of some sort, except for them. This speaks to the fact that they are in their false personas, and that each sees him/herself as Batman and Catwoman respectively. That is, at the masquerade dance, they were their false personas, just like everybody else.

Batman is the dominant personality, Bruce Wayne is the facade.

Bruce Wayne is definitely Batman, and Bruce definitely lives in Batman's attic. Batman ALWAYS comes first. It's who he is. He will give up any part of Bruce Wayne to make sure batman lives on. Bruce gives up his social and dating life. Spends billions on his vehicles and gadgets. And would even give up his human side itself to be a true hero who is unemotional and extremely effective. Bruce Wayne would die for Batman.

Just a billionaires son who inherits the business and grows up without parents and a thirst for revenge. Bruce Wayne became Batman to hide his identity to seek out the criminals of Gotham.

I believe because of the promises he has made to defend the defenseless and not be made weak by relationships we are actually watching Batman as Bruce Wayne from time to time.

Do i live in a house or do I live in America. Do I wear the shirt or does the shirt wear me. How many licks does it take to get to the center of a tootsie pop? The world may never know.

since batman has no powers is the first one u said

They are two souls living within the same body.