Neighboring rights and publishing rights (rights to the musical composition) are two completely separate businesses.
Neighboring rights have to do with sound recordings. Record labels and performing artists own the rights to sound recordings. Therefore, record labels and performing artists collect neighboring rights royalties.
Publishing rights have to do with musical compositions. Publishers and composers/songwriters own the rights to compositions. Therefore, composers/songwriters (and publishers/publishing administrators if the composer is signed with a publisher/publishing administrator) collect any publishing-related royalties (performance royalties, mechanical royalties, etc.).