I made this guide a few years ago and unfortunately due to changes in the iTunes policy, this app nor none similar are available in the App Store anymore.
To play YouTube music on iPod, iPhone, iPad or Android, MP3 is recommended. Now you can click on the "Audio" tab and select MP3 as.
With you can convert any Youtube's video into MP3 format and download It works on all iOS devices including iPhone, iPad and iPod touch. How To Directly Download Youtube Music Video To Mp3 On iPhone,iPad Mini,iPod Touch on (Any iOS)