Time shifting means copying for the purpose of listening to the recording at a better time, whereas space shifting is relevant here because the intended purpose is to allow the audio to be played in a different environment (i.e. not the YouTube player or app).

The more important thing is that space shifting to different media (digital, personal computer hard drive in this case) is legally treated differently. Though, to be honest, I'm not convinced that any personal recording protections would or should apply to the YouTube situation. It's just one way that the two differ.