They're not different stories nor continuing chapters, Beta 13 follows the restart of this project started back in the Beta 12. Basically, it's the same plot
(the Pokémon Adventures Manga's), but the tiles were changed, the conversations are a little bit more fledged-out and there's new content not available in the Beta 11.
It ends when you win the Kanto Region Pokémon League.
Do keep in mind if you intend to start a new savefile now, that you will have to start anew for the Beta 14
(which will include the Orange Islands and will be released in 2 or more parts) because the bag is going to be expanded and that automatically breaks previous savefiles.