Pretty simple explanation, they let the clearly copyright infringing content in to their walled-garden under safe harbour protections, but allowing download is contributory infringement with distribution and there's no hiding from that as they can clearly easily prevent it (as they have). Super hypocritical? Sure. Understandable? Also a yes.
It really chafes me that that somehow "looking at a video on a computer" managed to find itself legally distinct from "downloading it".
When you download the video, you are asserting more control over it than the legal system believes you should have.