I believe streaming vs downloading has slight implication differences. E.g if they choose to shut it off you can't stream anymore. Or force ads or view counts. Yet in downloaded form you are bypassing those things that have value
Example: In the US, the AHRA that makes personal space shifting digital audio legal (relevant to the submission) is fairly limited in the media that can be used for copying. Though I agree that they're being used for basically the same thing, computer hard drives aren't covered.
A relevant quote from Wikipedia [1] that sums up why I think this act falls short of its purpose: "In each case, the principal distinction between what is and is not covered by the AHRA is determined by whether or not the device is marketed or designed (or in the case of media, commonly used by consumers) to make audio recordings, not the device's capabilities."
[1] https://en.wikipedia.org/wiki/Audio_Home_Recording_Act#Digit...
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.