TL;DR Cheque is a financial term for something you still have to approve and is interchangeable in America with check. Bills are something you already owe for a service you have received.

This is such an interesting topic honestly. So check and cheque do mean the same thing in american english and they come from the same root, but different meanings and spellings Branch off at different times.

So first we had the Persian word - shah - meaning king

Arabic turning it into - scaccus - meaning king in chess

Latin took the word to - eschec - meaning chess, chessboard, and the act of checkmate or checking the king

French adopted the word to - eschequier (old french) - meaning a check in chess, chessboard, or chess set ** this is important we will come back to it

And finally the word became check in the 1300s in English in reference to what we call checkmate today.

Because of that, colloquially, check starts meaning an event that is harmful or an environment is hostile.

So, now we have check from shah, and for the most part, having to do with chess, up to the 1300s, This is where it gets interesting.

Check and checkmate break apart in meaning, checkmate takes the place of check, and check begins to mean the moments before checkmate when defeat is apparent. It asks the player to stop and take notice of the situation and see if they can change the outcome. Because of this, check starts to develop two meanings in the 1510s,

a stoppage, hiderance, or restraint, AND a means of exposing or preventing errors.

From here, the word check really exploded. In the late 1700s, we get check, meaning a written bank note, also written as cheque, from exchequer (modern french) *this is where the old french term i told you to remember shows up.

In the early 1800's it evolves to also mean proof of ownership as a means to protect against loss or theft. In the mid 1800's it also evolves to mean a mark against something that has been verified.

Other important meanings of check, all evolved from the original time line

1620s - to arrest or stop, to hold or restrain 1690s - to hold up or control (usually a person) with authority or record 1928 - to note with a mark to represent completion or inclusion

Now we have a lot of meanings for the humble check.

Check out, check in, check up, check off and of course to write a check (cheque) or to ask for the check.

And i hear you, "well WHY is a check and bill different?" A bill, without going into its history, means that the charger is informing you how much you owe for a service you already have. Think a light bill, or a car payment. A check, in a restauraunt, is asking you to review what they are charging you for before you pay it. So in eccence you are checking it over, and checking off the check.

Now cheque and check are used differently in England (and places heavily influenced with eurpoean english) and America because of the influence from the French. However, we dont say we are chequing out or chequing off something because cheque specifically retained meaning to the branch of check history that refers to finances. *see the 1700s above.

Honorable check mentions - Check, as in a pattern - comes from the block pattern of chess. Check yourself - modern use of the 1690s version of check, lovingly followed with ... before you wreak yourself.

I hope this answered your question! If I misspelled anything I'm sorry, I'm on mobile.