Boiled down to its most basic definition, style refers to the way in which an author uses language in order to create the mood and meaning of a text. This could include his or her word choice, diction, syntax, repetition, any figurative language (such as metaphor, similes, personification, apostrophe, metonymy, synecdoche, and so forth), etc. Style mainly refers to what kind of language a writer is using, and it is used in discussions of literature.
Stylistics refers to a study of the devices used in language that affect one's interpretation of a text. One might investigate, say, the particular rhetorical devices used by an author as well as why he or she would have chosen a specific stylistic device for a specific purpose.