If the story was linking directly to the "book, TV show, movie, video game, album, comic book, etc", and the link only worked for some people while others randomly got a login request or similar, you'd also see the top comment being a link to an archived version which avoids the login screen. That is: the main difference is that the archive link has the exact same content as the link submitted in the story, only bypassing the login screen that some people see. And the only reason the archive site has the content is that it didn't get the login screen; if everyone always got the login screen, what you would see on the archive site would be the same login screen.