Why a buyer outside China should care
You do not need a fapiao for your own accounts. You are not claiming Chinese input tax and your auditor will not ask for one.
It matters as a signal. A fapiao is issued through the tax system against the company's registration, so a supplier who issues one has booked the sale. A supplier who asks to be paid without issuing one — or who offers a discount for going without — is proposing that the transaction not appear in their records.
What that has to do with your money
An order that is not on the factory's books is an order their own management may not know about. If the relationship goes wrong, the paper trail you would rely on is the paper trail that was deliberately not created.
It also tends to travel with the other request worth refusing: payment to a personal account, or to a Hong Kong company with a different name. The two together are not a tax arrangement. They are an order running outside the company that would have to answer for it.
What to ask, and when
Ask early, in the same conversation as payment terms — not after a dispute. "Will you issue a fapiao for this order?" is a normal commercial question and the answer costs nothing.
A supplier who says yes has told you the sale is on their books. One who hesitates has told you something too.