I don't know what the actual answer is, but I am having the same issue. The page https://developer.apple.com/library/safari/documentation/AudioVideo/Conceptual/U sing_HTML5_Audio_Video/AudioandVideoTagBasics/AudioandVideoTagBasics.html
https://developer.apple.com/library/safari/documentation/AudioVideo/Conceptual/U sing_HTML5_Audio_Video/AudioandVideoTagBasics/AudioandVideoTagBasics.html
has some good info. And the page
https://developer.apple.com/library/safari/documentation/AudioVideo/Conceptual/U sing_HTML5_Audio_Video/Device-SpecificConsiderations/Device-SpecificConsideratio ns.html#//apple_ref/doc/uid/TP40009523-CH5-SW1
states
"Optimization for Small Screens
Currently, Safari optimizes video presentation for the smaller screen on iPhone or iPod touch by playing video using the full screen—video controls appear when the screen is touched, and the video is scaled to fit the screen in portrait or landscape mode. Video is not presented within the webpage. The height and width attributes affect only the space allotted on the webpage, and the controls attribute is ignored. This is true only for Safari on devices with small screens. On Mac OS X, Windows, and iPad, Safari plays video inline, embedded in the webpage." -
Which, I believe, means on the iPad using Safari and embedded videos, that the video plays at the resolution of the initial window (determined by the programmer), and does not naturally play on the YoutTube player, or at a higher res if you enlarge the video. And since you can't request a higher resolution from YouTube bc there is no tool bar option, you get the res you get regardless of your communiation speed.
This seems to be the "problem" that we're encountering - that its a "feature" specific to Safari on the iPad (not ipod, not iPhone, not desktop). Why they do this, I dunno. Its terrible. In my case, the video plays at a higher res (and better) on my smaller iPhone than does on my iPad mini (3/2014) with Retina. Go figure.
I came upon this post, hoping to find the answer and a work around. Yeah, its a major Apple issue but specific to the embedded video viewed through Safari on the ipad. This probably affects the quality of about 30% of all videos seen on websites with ipads. (10 million a day? - so no big deal right?)
I would love to be wrong on this, better yet, see an immediate fix.
Enjoy