Is there an advantage to using nitrogen and will it save me money in the long run?

With nitrogen, your tire pressures will remain more constant, saving you a small amount in fuel and tire-maintenance costs. There will be less moisture inside your tires, meaning less corrosion on your wheels, saving you money on costly refubishments or bead leaks which can lead to tyre damage. You will not be able to feel any difference in the ride or handling or braking, unless your tire pressures were seriously out of spec and changing to nitrogen brought them back to the correct number, but are far more likely to stay at the correct number after the nitrogen has been installed.