The biggest reason your payoff amount is higher is daily accrued interest.
Under federal law, the servicer must generally send you a payoff statement within seven business days of your request, subject to a few exceptions. (12 C.F.R. § 1026.36.)
A bank confirmation letter serves to assure all concerned parties in a business transaction that the bank's customer (the borrower) has, or has available, the necessary financial resources to conclude the transaction.
Instead, you have to get a 10-day payoff estimate from your current lender, which includes the amount you owe, as well as any interest that might accrue on the principal balance in the next 10 days.