You're right that you own the copyright (in the US) from the time you create the work. It's proving it that can be difficult and costly (court fees, lawyers, etc.) should you ever have to prove that you created the work and when, if you don't register your copyright.

Some people ask why you should bother to register (protect) your copyright it it's a song (or other work) that you don't aim or expect to make any money from. One reason is that someone else may later claim the copyright on all or part of your work - e.g., the melody, parts of the lyric, etc. This can happen because they've heard your song or poem and deliberately 'stolen' it. It can happen because they saw or heard it and forgot where. Or it can happen purely by accident, when 2 or more people independently come up with the same or very similar melodies or lyrics.   But if someone else does register a copyright on the same or very similar tune or lyrics, you can be sued for copyright infringement for performing, publishing or recording your own work, if you can't prove that you created the work at an earlier date.

This sort of thing has happened fairly often with folk music, parody lyrics, etc.    Sometimes it's whoever officially registers the copyright who gets to claim exclusive rights to the song.