When you play in the browser, you are playing on youtube. when you download, you can play wherever you want. This is similar to saying I'll go record a movie from a cinema hall and watch it later at home since I paid for it.
No it's just a simple, direct assessment of a situation. There is no technical difference between "streaming" or "downloading" the bytes. In each case, the exact same thing happens, the numbers are copied to the local machine. The distinctions are a legal fiction that doesn't actually have a reflection in the real reality, and as most imposed fictions end up, there is very little respect for the idea on the ground.
True, but at the end of streaming, the provider/browser clears the cache. The legality of it is debatable, but downloading copyright material is certainly not legal.
If someone visits me at home and uses my coffee-maker the action is same as taking it to his home and using, but only one of them is acceptable.
I did not say breaking in. That would be equivalent to hacking. If i'm home and I let my friend use my coffee-maker, that is analogous to youtube giving you a license to watch a video.
When you watch a video on Youtube, you are essentially given a license to watch that content on their site. They show some ads and make some money. When you download, you are violating that license.
Depends on the jurisdiction. Some make a explicit difference, e.g. German law speaks of (rough translation of the critical bits by myself) "temporary reproductions that are [...] integral and necessary part of a technical way of a) transmitting through a network [...]" which in practice means that streaming is ok in cases where explicit saving is not. For private use, downloading from not clearly non-legal streaming sources is fine though (again, in Germany)