Invoice are recognized by law，receipt is a proof of payment.

Cherry was correct. To answer your 2nd question, both can be deemed as a proof of payment in daily life. But invoices are more formal in contrast to receipts. Thus you may need an invoice in China when you need to get reimbursement from a company. Additionally, only entities can issue invoices in China for taxation reasons while natural persons can only provide a receipt, if you need one.