You get to be the Gotdamned Batman!
But seriously, the stories are linear in that you have to accomplish X before doing Y, but in that regard GTA is Linear, so is Assassins Creed, and a ton of other games. You basically get a Sandbox environment and then you choose what to do and when to advance the story.
For the real Batman is amazing experience, you should try it out on "Easy" mode. Anything else is annoying once you get to the badguys with guns sections.
I didn't enjoy Asylum nearly as much as City. Origins was fucking horrible to be honest.
City is just badassed all around