It differs in countless ways, most of them poorly understood. It generally deals with individuals rather than analyses or averages, with motives and reasons rather than movements and causes. It has a point of view rather than a "view" from nowhere. As the writer's maxim says, it shows rather than tells, contains dialogue rather than only declarative sentences, relies on context rather than raw data alone, is open-ended and metaphorical rather than determinate and literal, is tied to a particular time rather than being timeless, and deals with emotions rather than impersonal facts. Furthermore, narrative logic must deal with the notion of "common knowledge," whereby two or more people know something, know that the others know it, know that the others know that others know, and so on. In short, narrative logic is much harder than mathematical logic.
In everyday "story logic," how "we," the story-tellers, characterize an event or person is crucial. If a man touches his hand to his eyebrow, for example, we may see this as an indication he has a headache. We may also see the gesture as a signal from a baseball coach to the batter. Then again, we may infer that the man is trying to hide his anxiety by appearing nonchalant, that it is simply a habit of his, that he is worried about getting dust in his eye, or indefinitely many other things depending on indefinitely many perspectives we might have and on the indefinitely many human contexts in which we might find ourselves. A similar open-endedness characterizes the use of probability and statistics in surveys and studies.
Furthermore, unlike mathematical logic, story logic does not allow for substitutions. In mathematical contexts, for example, the number 3 can always be substituted for the square root of 9 or the largest whole number smaller than the constant without affecting the truth of the statement in which it appears. By contrast, although Lois Lane knows that Superman can fly, and even though Superman equals Clark Kent, the substitution of one for the other can't be made. Oedipus is attracted to the woman Jocasta, not to the extensionally equivalent person who is his mother. In the impersonal realm of mathematics, one's ignorance or one's attitude toward some entity does not affect the validity of a proof involving it or the allowability of substituting equals for equals.