A couple of advantages to Nitrogen but none that would significantly improve anyone's passenger tire performance.......The nitrogen is pure and free of any moisture content which reduces oxidation and the resulting deterioration but more to the point, it reduces the rate of heat build-up and makes the heat building far more predictable. This has no real effect on YOUR car but it does in racing. F1, Indycars, and NASCAR have used nitrogen for years though for just that reason.
Keeping your tires properly inflated, rotated, and balanced with your car correstly aligned will do FAR more to increase the life of your tires and your fuel mileage.