I do a lot of kart racing and the teams with real budgets use Nitrogen in their tyres (we don't, we're a poor team!). When you use air, you also get the moisture in the air into the tyre and the expansion of the moisture is very sensitive to temperature. This means that the pressure of the tyre is also very sensitive to the tyre's temperature.
In racing, the tyre pressure/temperature balance is critical to the correct functioning of the tyre - as little as a single PSI can add or remove significant time to your laptimes, so it's a balance that has to got right.
Because the moisture in the tyre causes the pressure to fluctuate with tyre temperature tyre pressures have to be set once the tyre is up to working temperature which on some makes can take twenty minutes (and this really needs to be done every few hours during testing). When you use Nitrogen, tyre temperature has a geatly reduced effect upon tyre pressure because you don't get the moisture in the tyre. The immediate benefit is that the pressure you set on a cold tyre is pretty much the pressure you get on a hot one. This means setup of the pressures is much easier, but it also means you've removed one of the variables (pressure) from the tyre pressure/temperature equation as you're driving which makes it easier to keep the tyre conditions consistant.
There may be other advantages to using Nitrogen but air is 78% Nitrogen anyway, so I would expect that the main advantage gleaned in the racing world is the moisture content issue.
Nitrogen in road cars? If it's free, why not. But you'd never notice the pressure consitancy benefits on the road. Maybe on the track, if your track driving was good enough.
Jon