They share a common Latin root, publicus, referring to the people. Both "publicize" and "publish" mean " make public" but in different senses. "Publish" is connected with the process of creating and spreading information. Traditionally, this was done in print, and even today it refers to written words. Book, articles, and oapers are published. "Publicize" is connected with spreading information, but generally not creating it, and can be written or spoken. Consumer products and events are publicized. In your example, the word is publicized. "The case was publicized in local newspapers" But if you refer to the case in a different way, you can use "published" "The story about the case was published in the local newspaper."

No, publish does not derive from public. However, they share a common root word or at least very similar words in Latin, which is where the words come from. You can say both it was published and it was publicized. Published means it was, at least once, printed in words. Publicized means it was "made known", but this could be done through any means, such as commercials, publishing, or even just telling all of your friends.