Actually, the biggest difference would be the supplier identification. A payment is not recorded to any supplier’s account, because supplier’s are subsidiary ledgers of Accounts payable. The Payee field for a payment is not linked and is not a subsidiary ledger of anything.
Doing what you suggest is not a matter of adding a button. It would require redesigning the program to link cash payments and receipts to suppliers and customers. This has been in the ideas category since the beginning: Improvement to payment and receipt forms.