From all the endless documents and articles i've read... i "believe" there is no cross functionality (i mean gaming, chat etc... play or talk with One users on the 360 and vice versa) whatsoever between the 360 and One for a few reasons.
The 360's operating architecture is fundamentally different from the One. The 360 runs on a PowerPC platform while the One runs on an X86. The Xbox One and PS4 are now PC architecture... great for us gamers! Bad for trying to play old games on next gen... this is the major reason we can't on the One. The PS4 can only b/c it live streams the PS3 games with the Gaikai servers.
Xbox live chat on the 360 is all analog while the One's actually uses the integrated Skype technology... yes, on the Xbox One, live chat will be powered by Skype and will (supposedly) sound just as nice! Which is fucking amazing!
I think these are two of the many reasons they are not compatible. But who knows... there has only been rumor and speculation so far... they might end up playing nice together... although i doubt it.
And yes, Live will be the same.. you buy the general Live card and can use it for a subscription with your One or 360... or both, depending on which you have plugged in at the time with your GT on.