My understanding is that when filled with nitrogen the tires heat up less and are more stable. As already mentioned because it is Dry Air and does not easily absorb moisture this avoids the added weight of water droplets and possible imbalance caused by any significant build-up.

I do know that all the serious Race Teams use Nitrogen in their tires, so there must be something to this.

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1991 M-B 560SEL Arctic White/Grey 99,000 Miles
1987 M-B 300SDL Ivory/Palomino 229,000 Miles (sold but never forgotten)
2006 Volvo XC70 Blue/Beige
1999 Porsche Boxster Arena Red/Savanna Beige
1986 Porsche 928S Goldweiss/Brown