Qumahlin nailed it on the head. CD-RWs cannot be finalized like a CD-R. However, there are a few portable CD players that can play CD-RWs now, thus allowing you to burn a mix album or whatever, then at a later time you can erase it and do another one.
I've never used a portable MP3 player, however, why would it require a finalized CD? That's kind of backward to me. If you can upload to the player .mp3 files and it can play it, why can't it just read the CD the same way?