One disadvantage of using a large, commercial bank can easily be seen if you're trying to get a loan. Unlike a local bank, or a relatively small bank, a larger, commercial bank will have to put a loan through several different departments. Beyond that, you may have to have dozens of people sign off on a single loan. This can lead to many more people getting involved in saying yes or no to your loan, and it may lead to a lot more negotiation than you were hoping to conduct. This is especially true for a simple, relatively straightforward home or business loan.
Another downside of using commercial banks is that they have very rigid standards more often than not. All banks have to follow the financial laws put forth by the U.S. government, but commercial banks may treat their own, additional rules as if they're set in stone. Again, this is most often seen in the loan process. Commercial banks, due to their size and the sheer volume of the market that they command, are often less likely to make concessions to customers. This can lead to a very "my way or the highway" attitude from a commercial bank.
One of the biggest concerns that a person has with their bank is whether or not their money is insured. If you put $10,000 in a savings account, you want to be sure that money will be available, regardless of what expenses your bank has to deal with. This is why the U.S. government created FDIC insurance, which insures up to $100,000 worth of money (though it's more than $200,000 until 2013) per depositor so that those depositors can have faith in the bank. Many commercial banks, because of how they choose to run their business, forgo this government insurance and instead offer private insurance. This private insurance isn't as reliable as the government's insurance, and it makes many depositors nervous to be out from under the government's protection.