The fax machine quickly overtook telex as the main method for sending and receiving letters and notes between companies. At the time - about 20 years ago - Telex was the only form of large scale text-based, instant communication, and was the only legally binding method of electronic communication.
The fax machine meant that in addition to ASCII text, you could send letters, letter heading, graphics, and it could be used be anybody who could dial a telephone. That made a big difference to how, why and when it was used.
Although email has replaced the large scale generic text based communication, faxes still rule when it comes to sending hand written notes, or when you need to send a form that can be filled in and sent back. I also know of companies that still use it as the prime method for proofs and as the main method for accepting orders, even though the orders are often computer generated and then printed and faxes in each case.