Invoice and bill usually name the same document from opposite sides of a transaction. The supplier issues an invoice; the buyer, and most accounting software aimed at buyers, calls the thing that arrives a bill. There is no reliable technical distinction, and the confusion mostly matters when software uses the two words for different records.
The everyday usage
A seller raises an invoice and records it as a receivable. A buyer receives it and records it as a bill, or as a payable. Some accounting packages use bill specifically for what you owe and invoice for what you are owed, which is a helpful convention rather than a definition, and other packages do the opposite.
Where the words diverge slightly
Bill is used more loosely in consumer contexts, such as a utility bill or a restaurant bill, where no purchase order and no verification are involved. Invoice carries a stronger business connotation of a formal claim with terms, a number and a due date. That is a difference in register rather than in substance.
Why it matters in practice
Only where your software distinguishes them, and then only because people search the wrong list. If your system calls incoming documents bills and your team calls them invoices, agree one word internally. Consistent vocabulary is worth more than being right about which one the dictionary prefers.
Questions people ask about invoice vs bill
Is there a legal difference?
Requirements attach to what the document contains rather than to what it is called, and they vary by jurisdiction. Check your own obligations if it matters.
Which term should we use internally?
Whichever your system uses, consistently. The cost of two vocabularies is people searching the wrong place.
What about a statement?
That is genuinely different: a statement lists outstanding invoices rather than being a claim for a specific supply, and should never be paid from.