Consider this from the point of view of an accountant. When accounting for a firm's financial transactions, accountants record revenue and expenses through accrual accounting, as opposed to recording cash receipts and payments via cash accounting.
For example, if you purchase A for $X on Jan 1, you receive an invoice for $X and the firm notes that it has generated $X of revenue. More importantly, the firm can provide you with some credit or terms of payment that allow you to pay up later, but this $X is recorded as revenue upon sale. Eventually, you'll have to actually pay them $X and you'll receive a receipt after doing so, but most firms trust that you'll pay up eventually because of the going concern assumption; firms assume that you're financially stable enough to pay your dues when they are due.
This line of credit is why receipts and invoices are separate documents and both should be kept. An invoice without a matching receipt means that somewhere out there, someone owes you money for something that has been purchased and not paid for.
Conversely, a receipt without a matching invoice means that someone has given you money but you don't know what you've given them. This generally means that something has gone wrong internally (the exact phrasing used by accountants is internal controls). For example, the company might have lost track of some inventory or screwed up its cash receipts. This might be a minor mistake, but internal controls are also paramount in ensuring that people aren't embezzling from the firm or stealing money from customers.
If a firm is legally required to produce audited financial statements, the firm's internal controls will be examined by an auditor. This includes tallying up receipts, calculating the amount of money owed by customers (i.e., the accounts receivable) and how long it takes to receive money on average, matching inventory records to physical inventory, and so on. I've worked with auditors, and while we don't require every single receipt to be present, we do expect most of the paperwork to be filed and kept on hand.