Does Apple TV work better with a 4K television?

Apple TV 4K will work fine with an HD TV, but the resolution will be what both devices (and the content) support. As such, some features will not be available if all of your setup doesn’t support those.

Please elaborate if your meant something different by ‘performance’.

Sure, because it can display 4K content assuming that is what the content being shown is streamed in 4K. Remember it is streaming which means it needs a excellent network signal. At a minimum you will need a network that supports 25 mbs where the Apple TV is.