We are all creators. In a way, our steps, our thoughts, and our melodious voices are marvellous compositions of the finest music.
A song author can dictate music to others. That means, a song author can play music, record it, perform it etc., yet a song author may not have the skill to bring the music to paper.
A songwriter can write songs down. It's in the name itself: "song" "writer." As a writer writes, so does a songwriter write songs.
A composer writes many types of music for various types of groups: band, orchestra, chamber orchestra, soloist, etc.