In mathematics, “difference” is the result of subtraction. “The difference between 5 and 2 is 3” is another way of saying 5-2=3. In your example, the speaker is using this same definition of “difference.”

Example: I booked a flight on November 15th for $100. Now I want to change the date of the flight to December 15th. The price of the flight on this new date may be different to the original price I paid. If the new flight costs only $75, then the airline should refund me the difference ($25). If the new flight costs $125, then I need to pay the airline the difference ($25).

In this sentence, "difference" refers to the remaining amount or discrepancy in money that needs to be refunded. For example, if a customer paid more than necessary for a product or service, the "difference" is the extra amount they overpaid, which will be refunded to them. So, here, "refund the difference" means giving back the amount of money that was overpaid.