Payoff amount and current balance are related but not equivalent terms.
Current balance means the amount you owe according to your statement. The next day, you will owe more. In other words, if you are trying to pay off a credit card and the statement says your balance is $514, you may not be able to bring your balance to zero and satisfy the debt by writing a check for $514. Instead, you would need to contact your lender to find out your payoff amount.
Payoff amount is how much you would have to pay to satisfy the debt. It is not the same amount as the current balance on your statement—at least not for long.
The difference between the current balance according to your statement and the payoff amount is crucial when you are ready to pay off your debt.