What defines an apocalyptic story?
The easy answer for me is
The End of the World as We Know It
I think it's the "as we know it" part that is most important. Apocalyptic stories start from a point of the world that we know, that we are familiar with. I think it would be pretty hard to appreciate an apocalypse in a world utterly unfamiliar to us, since we'd have no "before" to compare to.
Also, the apocalyptic event affects the whole of society. After all, we all go through experiences that change our personal world as we know it. Marriage, children, school, jobs, divorce, death, etc. But those don't make an apocalyptic story. Still, as I think about it, maybe it is our own personal apocalypses that make world-changing apocalyptic stories so appealing to us.
Be sure to check out the responses from the other panelists!