The main difference is how they affect your books upon recording
Customer Deposit allows you to record Customer advance payments. These payments are recorded in your general ledger as a liability until the goods or services are actually delivered and do not affect the customer’s accounts receivable balance. Then, you can apply a customer deposit against an invoice when the order is complete.
When the deposit is applied against the invoice, the liability is canceled out. It creates a Deposit Application record that impacts the Accounts Receivable.
On the other hand, Customer Payments are directly credited to Accounts Receivable when created.