A department store is a large store (shop) with different departments (sections) which sell different products such as shoes, clothes, jewelry, cosmetics, books, tools. In North America, it was a predecessor to the shopping mall. It provided one-stop shopping.
A shopping mall is collection of independent stores within a large (often enormous) building. A shopping mall often has many shoe stores, clothing stores, and jewelry stores. With the advent of online shopping, many North American malls have closed.