ok your link and batboy's link I can play in 4k in IE (firefox the option isnt there in menu).
Both no stuttering but I do get a momentary glitch after going full screen.
Basically default youtube behaviour unless unticked hardware acceleration box is this.
Software video rendering, hardware video decoding in window
Hardware video rendering and decoding full screen.
So try full screen if window is stuttering and it may well be smoother.
Mine is ok in both tho, maxes at about 20% gpu usage and not much cpu usage.
I have a haswell 4670k overclocked to 4.3ghz and gtx 760.
On the quality, looks no different to 1080p videos to me.