Fax is short for “facsimile” when used in the context of electronically transmitting a document between fax machines.
The official definition of “facsimile” is a copy or reproduction of an item (typically a work of art or a document).
A reproduced work of art is called a facsimile, as is a reproduced map. Neither of these are called faxes.
A document electronically transmitted between fax machines is called a fax. It can also be called a facsimile.
Origin of the word
Traced back to 1938, the term “telefacsimile” was one of many inventions at the time that added “tele” to a traditional word (such as television or telephone).
Later shortened to “telefax,” (that appears to have originated in the early 1940s), “telefax” was likely shortened to “fax” the same way that “telephone” has been shortened to “phone.”
Sources and more information:
Wikipedia – Fax
Adams on Contract Drafting – A Gripping Issue of Notice-Provision Technology: “Telecopier,” “Facsimile,” or “Fax”?