The terms "motif" and "motive" mean very different things in English although they both derive from the Latin verb "movere" (to move).
Motif comes into English via French and means a theme or recurring idea in a literary work. It is normally an idea which recurs and has some sort of symbolic significance for the story. In music, a motif is a recurring musical idea or melodic fragment.
Motive arrives in English via the late Latin "motivus" and means motivation or something that impels one to act a certain way. In analysing plot in narratives, one often analyses what motives caused characters to act a certain way.