I understand that it'll download updates in that state, thats what it specifically mentions in the settings (Xbox will download updates in standby) but it mentions nothing about full games. I've got competing answers here now
Well I know that when downloading Forza 5 I turned mine off (No light on the console, no HDMI signal, etc.) and it still downloaded the game (50% in 10 hours woo Australian internet!). If you connect it to a switch you'll see the connection go from gigabit (When it's on) to 100Mbps (in "very low power mode"), and no light if it's actually turned off.
Interestingly, the PS4 never changes the link speed, "turned off" and it keeps the port at gigabit speed.