Basically the difference is that with owning your masters gives you the rights to distribute your own music. You alone makes the decision on what service can host your music.I believe it close to being one in the same this what I found on the subject
What is the Difference Between Master Recordings and Music Publishing?
Publishing means you get paid off people using your music. For instance the owners of your masters can allow a third party to use your song for a movie, but if you own the publishing, you get paid for them using your song.
Seems like you'd make more money from publishing but would have more control of your music with masters.
That's what I got from this article.