Music is its own language. The notes and rhythms resonate to us musicians the way words and grammatical structure do to an author or poet.
To an experienced musician–composer and songwriter–the musical language becomes a second language to us. To someone who is just starting out as a songwriter, that musical language will seem foreign to them.
I have seen very talented musicians that play by ear, and do it very well. Why is it important for songwriters to learn to read and write music? Why isn't being able to play by ear good enough?
Let's say you're at your instrument, and you come up with a catchy melody. I'm talking Grammy award winning stuff! I'm very forgetful that I write everything down. As soon as you come up with a song idea, the easiest way not to forget it is to write it down–a basic notation if it's piano or tab it if you're a guitarist.
As your songwriting gets better your compositions will become more complex. If you only play by ear you'll have to play your songs many, many times just to remember how it goes. And you may not play it the same way each time. Being able to write down your songs eliminates such a long and tedious chore of repetition until memorized.
Understanding the music language, with some basic knowledge of music theory will go a long way. Such a valuable asset to have at your disposal. Playing your instrument will no longer look like you're pressing random keys on the piano, or plucking random strings on the guitar. It's a satisfying feeling knowing what you're playing.
This will also help in arranging your songs. Maybe a chord progression or melody will sound better in the verse rather than the hook. When it's written down you can visualize like a panoramic view of the entire song all at once, and make necessary changes as you see fit. Changes that could be the difference in whether or not your audience likes the song.
Having an "ear" for music is one great tool to have as a songwriter. When you add reading and especially writing capabilities it will make you a well-rounded songwriter.