I'm not a millionaire but I used to work in a bank and I know we would regularly waive fees or give perks to people with higher balances. It always pissed me off because I felt that the person who was making $7/hr needed it more than the rich person but I didn't get to choose.
I experience this as well. If you keep a balance of about $10k in a Bank of America account, you qualify for a special account type that has pretty much no fees. That's not terribly large if you keep an emergency fund type balance in there. It does make sense in that they actually profit a lot more off of you than someone who keeps $50 in their account.
It's not that they profit "more", it's that they might actually profit at all. The $50 dollar account is almost guaranteed to lose them money every single month, just due to things like ATM operations, keeping branches open, mailing their statements at $1+ a pop, etc.
If you have a high income or large amount of investable assets, you may qualify for your bank's version of "Private Banking". There will usually be a local person you specifically deal with that has more authority than a normal branch banker, your accounts will have less fees, you might get free checks, you might have access to custom loan programs that "regular" customers don't have access to. There could be a number of other benefits, and the more money you have the more access you get. Thresholds for qualifying will vary by institution and geographic area, but think $100-200k/year income and/or $1-2million investable assets.
As far as anything that might actually benefit you, they may take you to lunch or invite you to exclusive events with other customers of theirs who you can network with. Don't expect them to bend over backwards for you if you don't actually do any business with them, though :).
