I'd like to add to David's accepted answer:
Despite the apology of inventor of the web I think the double-slash syntax served an important purpose: to visually stand out. Double-slashes allowed easy visual distinction of URLs in a text without hyperlinks. When you saw double slashes you immediately thought it could be entered in a browser window, similar to how you thought a text containing
@ could be used to send an email. It was especially crucial during the transition phase to web where protocols of that era (ftp, telnet, gopher) had their own weird notion to represent server addresses or resource paths, rarely both. Most of the problems associated with double-slashes would still exist, because double slashes are the least cryptical part of a URL, think about port numbers, percent encoding and case-sensitivity. But having a URL like http:something.com could easily be confused with my example here:something.com. Look at http:// on the other hand, how it shines like a diamond. Double-slashes have been an important part of Web symbolism and I believe it accelerated it's adoption rate too, even if it was unintentional.
They also might have made AmigaOS's job easier to differentiate between file names and URLs since AmigaOS used the file path syntax