I've heard various supposed benefits to running nitrogen in your tires, some more plausible than others, but this is a new one on me. How exactly does nitrogen prevent temperature from affecting tire pressure (or reduce the effect, even).
Before you answer, consider that almost 80% of our atmosphere is nitrogen so any benefit from nitrogen is already 80% present in regular compressed air...
Nitrogen is less prone to expansion and contraction with tempurature variations. We started putting it in all the new cars we sell because customers were coming in with their tire pressure monitoring lights every other day. It was driving them and us crazy. The problems have dropped off to almost nothing since we started putting the nitrogen in. It's also supposed be less corrosive on aluminum rims, but I don't have any info to back that claim up, that's just what they told us in training. As far as claims that it increases fuel economy, those claims are from people who were running low tire pressure to begin with. We had a lady complaining (imagine that, a woman complaining) about her fuel economy, we went to put the nitrogen in and her most inflated tire was at 24 lbs. Look for a deal on it. We charge $40 regular price, $30 when it's the service managers "special".
According to NASA's studies, higher pressure is better against hydroplaning. I believe it correlates with 1/sqrt(pressure), so it's a diminishing returns thing. However, lowering the pressure would make it more likely to hydroplane, not less.
Snow may very well be another story. I can't comment on that.