There is one potential difference: a Xbox console would be running Windows 10 and use standard hardware.
I think that all Microsoft would do is add 10-foot interface components to go with the tablet and desktop interface components, then treat all consoles as a standard PC with HTPC elements. I believe gaming enthusiasts would be able to build their own consoles from scratch - like when they make their HTPC - then just buy a Windows 10 license and plug in a controller. I can see Xbox features like chat becoming standard Windows 10 apps, and having games purchasable from the Windows Store.
This would allow for people to use PC sticks with the Xbox interface, and play Xbox games on systems that support it. I suppose Microsoft would set powerful minimum requirements for the consoles, and force companies to meet power and driver requirements to be certified as a Xbox console.
But it is doable, and would work far better than a bunch of similar black-box systems with mysterious and nearly identical innards.