I don't think D&D was designed as a story-telling tool, rather story was injected into a tactical game, but it does come out feeling like a 500-700+ page fantasy epic, or a long running TV show. (Or it has felt like we are starting one, I have never played a campaign to completion.) On the other hand when I have played some Powered by the Apocalypse, the feel is that of a 90-minute action movie.

In D&D, the general trend is towards stability. Combats are resolved, healing is plentiful and characters level up and grow stronger. In the Apocalypse Hacks I have played, you survive combat, live on with scares and there is a very real chance ending up in a worse situation then when you started. One is not better than the other (although I enjoy the lower time investment of the latter) but at the same time they conjure up very different feelings.