That's a very good question, Ismael. They mean more or less similar things, but here are some differences I can cite from daily living by giving you a couple of scenarios:
1a. You receive a box of merchandise you purchased that has been sent to you through the post office or an international carrier like Federal Express. You open up the box and find a paper in it listing all the contents of the box and the price per item. That's an invoice.
1b. You're a professional writer who writes on an ongoing basis for a company. You're paid by the hour and, once a month, you send a list to your supervisor of all the days you've worked that month and the hours you've worked on each day. He'll add up the number of hours and multiply that by your hourly rate of pay to determine how much you earned that month and how much money he should send you. That's another kind of invoice.
2a. You had your plumber fix a leak under the kitchen sink and you also had him install a new toilet in the bathroom. He hands you a bill when he's finished all the work. You write him a check after looking over the bill and seeing that it's accurate.
2b. You and your spouse have finished eating dinner in a restaurant. You tell the server that you'd like to leave, and a few minutes later he brings you the check. You look over the check, determine it's okay, and then give the server back the check along with your credit card so that you can pay the check (which is the special word we use for a bill in a restaurant).
I hope these scenarios have given you a clear idea of how we use these two nouns, Ismael.
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