Advertising: The public announcement of a persuasive message by an identified sponsor; the presentation or promotion by a firm of its products to its existing and potential customers.
Marketing: The systematic planning, implementation and control of a mix of business activities intended to bring together buyers and sellers for the mutually advantageous exchange or transfer of products.
In my opinion the difference is simple. Advertising is the things you normally do to actually catch the attention of your prospects. Think PPC ads, direct mail, telemarketing, flyers, etc. Marketing is more of how you actually position yourself in the eyes of your potential market. It is the process of communicating to your market, what you do, why you're different, and the niche that you so perfectly fill. Obviously there is going to be some overlap between the 2.
Advertising is about communicating the existence of of a product or maintaining brand value. It always follows marketing. Marketing is about finding connections between users and your products (who is using what -> to adapt advertising communication and allocate advertising resources efficiently).
Somehow, you could consider marketing as the strategy part and advertising the execution.