Imagine if a singer wrote a song and only he was able to sing it live to other people. Does such a thing exist?
IANAL, but...
Technically, Yes. Technically, every local band in every local bar is performing copyrighted material without the license to do so (unless it's their own original copyrighted material). However, it's almost impossible to enforce, completely non-cost-effective to enforce, and actually counter-productive (since it is in effect, free advertising) and might be considered fair-use(?). However, if you wrote a song, and someone else started performing it, and they were actually becoming better known for it than you (or they were performing it so badly it was ruining the song), it would be in your best interest (and your rights) to force them to stop performing it.
If one of these bands started recording versions of other bands songs, or selling recordings of live performances of other peoples songs, or started performing large venues with entrance fees specifically for hearing the music (as opposed to a cover charge at a bar, which is for more than just the band), then the copyright holder (whether the music company or the musicians themselves), could (and usually do) come down on the offender like a ton-of-bricks. And, they will win.
There are lot's of example of this happening. Boston is a good one. At one point in Boston's history, two former members of Boston started touring, playing predominantly Boston material and billing themselves as the "Boston" sound. Tom Scholz (Boston's founder) sued them (I believe for both Trademark and Copyright infringement) and won (but of course it took a lot of money, on both sides). Another good example is musicians sending C&D's to politicians for using their music in their campaigns without permission.
If someone wants to use someone elses material on an album (whether live or studio), or more and more commonly in commercials, they have to get permission, and the copyright holder is within their rights to charge a royalty fee (a percentage fee to essentially license the copyrighted material, with said license possibly restricting how and how-long the material may be used).
So basically, copyright is copyright, whether a text or music. They are both equally protected.