Both can mean "quality," but are used in different senses. "Calidad" can also have many other translations, and is by far the more common of the two words.
"Calidad" can mean quality as in the quality of a product or service (high quality clothing or a poor quality of education). It can also indicate in or under what condition someone participated in an event or transaction: "Mi amigo me representó en calidad de abogado." "Calidad" can also express the concept of the perceived "quality" of people in general: "Me gusta rodearme de personas de calidad."
"Cualidades" (pl.) refers to those qualities that are immanent in something, and distinguish it from everything else. It can also refer to qualities as positive attributes of someone's personality: "Esa mujer tiene muchas cualidades, y por eso ella me cae muy bien."