Bungie updated its Destiny Trial and Demo FAQ earlier today with information on an imminent change coming later this week. "The Destiny Demo will no longer be available starting June 23, 2016."
The developer describes the Destiny Demo as a "snapshot of Destiny at the release of Update 2.0.0, only available on PlayStation 3 and Xbox 360 consoles"—a sampler platter, in other words, meant to entice players into going whole hog on the retail version. The Demo allowed progress up to character-level 7; that progress could not be transferred to the full game.
Bungie's decision to pull the demo is one in a growing sequence of moves designed to herd players away from "legacy" consoles. Two weeks ago, the company announced that the Rise of Iron expansion due later this year would be exclusive to the PS4 and Xbox One versions of Destiny.