Two main reasons 'dry nitrogen' is used instead of straight compressed air, firstly it doesn't conduct heat as well as air and secondly it has no water vapour in it.
Both an advantage when the brakes get really hot and the heat starts to conduct into the wheel/tyre assembly.
The heat soak takes longer and the lack of water vapour ensures that steam is not generated in the hot wheel causing a much increased pressure.
Peak brake temps in a an airliner after a high speed rejected take off can exceed 900degC.
Concerning filling your car tyres with nitrogen, pretty much a waste of money IMO, what's already in there is 78.8% nitrogen anyway and very few car drivers will ever have peak brake temps much over a couple of hundred degrees if that.
Last time I had the tryes replaced on my car the tyre tech asked,
'Dya want nitrogen in 'em mate?"
I said,
"No thanks, I already run an 80% nitrogen mix"
"Oh" says he and wanders off.
Regards,
BH.