Purchase orders and invoices are the two ends of the same transaction and most small businesses only keep the second. That asymmetry is the whole problem: you have a complete record of what you were billed and no record at all of what you agreed, so every disagreement starts from a position where the supplier has a document and you have a memory.
Who writes it, and when
You write the purchase order, before anything is delivered, to say what you are buying. The supplier writes the invoice, after delivery, to ask to be paid. That is the whole distinction and everything else follows from it: one is a commitment you make, the other is a claim made against you, and they are written by different people at different moments for different purposes.
The number is what connects them
An invoice that quotes your order number can be compared to the order. One that does not has to be checked from memory, which in practice means it is paid. On the worked example order 1042 was agreed at $827.21, so an invoice quoting 1042 at $869 is a $42 discrepancy with a document behind it rather than a feeling that the bill looks high.
Matching, and how far a small business should take it
Large organisations match three ways: order, goods received note and invoice. A small business usually matches two, the order and the invoice, and checks receipt informally. That is a reasonable trade. What is not reasonable is matching zero ways, which is what happens when no order exists to match against.
What to do when they disagree
Query before paying, and query against the document rather than the recollection. A price difference is usually a discount not applied or a shipping charge nobody agreed; a quantity difference is usually a partial delivery invoiced in full. Both are visible in seconds with an order to compare against and effectively invisible without one.
Questions people ask about purchase orders and invoices
What is the difference between a purchase order and an invoice?
The buyer issues the purchase order before delivery to say what is being bought; the supplier issues the invoice after delivery to ask for payment. The order number on the invoice is what ties them together.
Which comes first?
The purchase order, always. An invoice arriving with no order behind it means the commitment was made without a record, which is the situation purchase orders exist to prevent.
Should we refuse invoices with no purchase order number?
Query them rather than refuse them, at least at first. One query is usually enough for a supplier to start quoting the number, and it also makes internal ordering without a requisition visible immediately.