There are several compelling reasons to inflate your tires with nitrogen instead of regular air.
First is that nitrogen is less likely to migrate through tire rubber than is oxygen, which means that your tire pressures will remain more stable over the long term. Tires filled with nitrogen rather than air also exhibit less pressure change with temperature swings.
Vehicles can also benefit from the more stable pressures. But there's more: Humidity (water) is a Bad Thing to have inside a tire. Water, present as a vapor or even as a liquid in a tire, causes more of a pressure change with temperature swings than dry air does. It also promotes corrosion of the steel or aluminum rim.
YOUR TIRE PRESSURES WILL REMAIN MORE CONSTANT, SAVING YOU A BIT IN FUEL AND TIRE-MAINTENANCE COSTS
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. 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 proper numbers.