We always used nitrogen for aircraft tires for several reasons. Pull the following info. from a web site and it might answer some questions.
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. Racers figured out pretty quickly that tires filled with nitrogen rather than air also exhibit less pressure change with temperature swings. That means more consistent inflation pressures during a race as the tires heat up. And when you're tweaking a race car's handling with half-psi changes, that's important.
Passenger cars 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.
How is water relevant to a nitrogen discussion? Any system that delivers pure nitrogen is also going to deliver dry nitrogen. Filling tires with nitrogen involves filling and purging several times in succession, serially diluting the concentration of oxygen in the tire. This will also remove any water.
It's certainly simple, although time-consuming, for a tire technician to fill and bleed tires. But most shops use a machine that not only generates almost pure nitrogen by straining the oxygen out of shop-compressed air, but will also automatically go through several purge cycles unattended. Some shops have been charging as much as $30 per tire for this service. I think that's too much. If you're buying a new tire, it should be far less. Still, the nitrogen generator, filling system and technician's time aren't free—the dealer is entitled to some return for that.
So, to answer your specific questions: 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.