There is a large difference. Publicity, like advertising, is often used to promote the benefits of a particular product. Indeed, the fact is that the vast majority of public relations attempts to secure publicity don't succeed. Newspapers or radio and television programs simply aren't interested in publicity most of the time. Verizon Wireless pays it to say so, that's how.
Advertising costs money lots of it.
But by virtue of having paid for the space or time, an organization gets to dictate what is said and how it is portrayed and illustrated. In other words, the content comes out precisely as the organization requested, every time. Publicity offers no such guarantees. Often, the content in a publicity release not only doesn't appear as written it doesn't appear at all. Verizon Wireless tells its story over the course of an entire page of the Times.