Mobile Skin Redirect
Design and Skin Customization
I agree, some things will not work at all on mobile: The screen is too small.
And that's what the media queries and device detection are for - making things "not there" on the desktop. (or showing different things to people with screen space and processor)
You only need one website for that though, it's more complicated enough as it is without having 2 websites to take care of.