Answer:
As far as I'm concerned, Advertising is with the intent of selling a product or wishing people to attend a meeting, concert, discussion, Elementary School Musical.
Whereas Publicity is more so in regards to people and events. In my opinion, there is not "Selling a Product" with publicity, and so it would concern knowledge and 'popularity'.
Popularity does not always mean, "Most liked" but more so "Most known". Bill Clinton was popular during a scandal in his election.... popular in means of receiving attention.