A motif: is a short musical idea, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition.
The differences: While a symbol might occur once in literature to signify an idea or an emotion, a motif can be an element or idea that repeats throughout that piece of literature. It is closely related to a theme but is more of a supporting role to the theme than a theme itself.
Why a writer might use a motif: A motif gives clues to theme or reinforces ideas an author wants to emphasize.