What is the purpose of asking me to keep the old bank account?

~~The reason is simply because if you are having it automatically deposited, it can take up to 30 days for the change to be effective in SSA system. If you closed the account prior to SSA updating their records, then your social security checks is returned to SSA and it can take a couple of months before it is corrected! I know someone this happened to, it was a nightmare for her.

So they are wanting you to keep that one open until you actually see the next check deposited to the correct bank account. Very wise advice, as soon as you know it is definitely in the correct bank, then you can close the old one.~~

I've heard of stories where accounts with auto deposit/withdrawal can incur fees if they're closed and a deposit/withdrawal is attempted. In your case, I would keep the account open just case:

A) Some idiot at Social Security fails to deposit your income into the new account. Instead, they continue to deposit into your old account. If your old account is closed, you won't have access to your money. Social Security will tell you it was deposited. It will take awhile for you and SS to clear things up all the while you don't have your money.

B) You might have some auto bill-pay linked to your old account that you're not thinking of at the moment. Let's use your utility bill just as an example. Well, when the utility company tries to automatically get your payment from your old account that you closed, they won't be able to. They'll assume you're not paying and might turn off your utilities. I know that sounds somewhat extreme.

Just keep your old account open for 30-60 days or longer. Check for any activity twice a week. When you're absolutely sure it's safe to close, go ahead and withdraw and give them the bird while telling them you're taking your business elsewhere.

Just to make sure the change goes through. You should not close the old account until you have received your next check into the new account.

If you switched banks the answer should be obvious...they want your business in the future. If you have an open account then there will be a constant reminder of the bank you used to belong to.