One you bank with and the other you don't. Thus, the one you bank with already has a lot of private information about you especially if you use any other services besides savings/checking. Thus, its much easier to open a trading account with the one you bank with...just make sure you understand their fee structure and reporting practice. Also, if you go the bank route...check and see if they allow "linking" between your different types of accounts. Its a plus if they do.
In contrast, a broker that doesn't do any banking...you're going to need to cough up a lot more information to open that trading account. Yeah, it seems like an overkill but if its worth it if you're like me and paranoid type about privacy and identity theft. I've been a victim once (discussed by me here many years ago) and I refuse to allow such to happen again. Thus, if my broker ask me what's my spouse favorite position just to open another account...I'm going to tell them.