Drake's recent win regarding the use of Jimmy Smith's statement inside of his 'Pound Cake' instrumental is a great example of the difference between Master and Publishing rights.
if music was like a book.

The book itself (pages, font, ink etc) is what the Record Labels own and sell to make money. it's the product or the file it travels in.

The story within the book, is the composition. it's intangible, but it has a value. This is what the Publisher owns, and they ensure the writer gets paid for every use or reference of the story.

In this case, Drake got permission to make a book with some of their pages, but he did not get permission to use some of the story. 'Fair Use' ruled that the piece of Jimmy Smith's story that was used did not add, subtract or change the value of Drake's story, despite the pages being there, so no permission in this case was required.