Difference between "invoice" and "bill"?

I am talking about something you should pay. "Invoice" here doesn't mean the proof of payment. Sometimes I am told to pay my "bill", and sometimes they may refer to the similar paper (physical or virtual) as "invoice". The only difference I can think of is that "invoice" seems to be something you pay before you use while "bill" is the opposite. But this distinction is not clear since my university also calls the tuition fee as invoice now (I am already studying when I pay the "invoice").

Could someone tell me the difference?

An invoice is a request for payment that contains information necessary to remit payment to the issuer. Normally that would include the issuer's name, address, terms of payment and if necessary an account number. The invoice also is given a number - a unique key identifier with respect to the issuer.

Later, this number can be referred to in future correspondence. A bill may or may not include any of the above, but an invoice usually will.

Rather, the distinction between these two words is more a matter of custom and the business in question being transacted. For example, a lawyer bills a client, for billable hours. In the abstract, "you should pay your bills on time." Work that is commissioned will usually generate an invoice.

Bill in the sense we're discussing is Anglo Saxon and dates from the 1400s, and invoice is French (envoyer, "dispatch") from the 1500s. You might make the case that invoices are attached to things that are delivered, but in practice it seems that various trades over time have selected one term or the other by tradition and custom.

An invoice is a formal written demand for payment: if you accept that it is valid then you should pay it in accordance with the terms of the contract, e.g. within 30 days. A bill is simply a list, so might contain details which are not necessary for the invoice, and may or may not be a demand for immediate payment. So you could imagine being told "Here is a bill showing everything you've ordered. I'll send you an invoice as soon as it's been delivered."