The big differences to me are that faxes don't need an "account" to send/receive files, the machines are simpler and cheaper, they have far fewer intermediary technical and user issues, and their network is way more reliable. If you depend on sending and receiving documents, faxes are light-years more reliable and less complicated than, say, e-mail.
Can you count the number of times an internet connection has gone down for a business, compared to the number of times the PSTN has gone down? Unless a truck takes out a utility pole, there's no contest. And the lack of obstructions for user access removes a whole slew of other issues.