PRS for Music is the UK's leading collection society, brining together two collection societies: the Mechanical-copyright Protection Society (MCPS) and the Performing Rights Society (PRS). PRS for Music is committed to protecting the value of music by ensuring that their members are represented, fighting music piracy along with tackling the changes that digital has brought to the industry.
Music Publishing is the business of protecting and promoting song copyrights and collecting the royalties that these copyrights generate. Think of a song as intellectual property (it's an original piece of art!), and music publishing as the business that makes sure that songwriters get paid a fair compensation when their intellectual property is used by companies. Those companies could be record labels (mechanical royalties), radio stations, bars, and restaurants (performance royalties), or film studios and advertising agencies (sync license fees), and many more.