Keep the old account open and forwarding for months, not days. Customers reply to old threads, suppliers keep outdated contact records, and some accounts send password resets to whichever address you signed up with. Check it regularly for anything forwarding missed.
Be careful with internet-provider addresses. Some providers close the mailbox when you cancel the internet service, and the address can't be recovered afterwards. Finish the transition, and copy any old mail you need, before you cancel. If your business still relies on a mailbox that's stopped receiving after a move, fix that before anything else.