He is stubborn.
Why is he stubborn?
He is being stubborn.
Why is he being stubborn?
The first one means he's stubborn in general, not just now.
The second one means he's currently being stubborn, probably about something in particular. Eg. if he doesn't want to do something, you could ask "why on earth are you being so stubborn?".
If you ask "why are you so stubborn", you imply that he's stubborn not only now but in other occasions too.
FRC