Tumblrponies that are in "Story blogs" may or may not have an ask box, but do not particularly use it to drive the story. The blog plays out like a webcomic, with only moderate user interaction (if that).
Obviously, almost all blogs will have some form of story. This category is for blogs that use the author's story as the sole driving force, and will stick firmly to hitting predefined events. In particular, RP blogs should generally not be labelled as story blogs, as the stories they create are highly organic.