As someone once said...
Nitrogen has been proven to reduce those nasty deposits of cash in your wallet...!
Otherwise, not much benefit to the average motorist....save your $ and buy a decent tire gauge and a small home air compressor.
But, someone will say " ...airplanes and racecars use nitrogen...! "
Yes, airplanes with retractable landing gear are required to use nitrogen, as it is less likely to explode in cases where a tire / brake assembly gets superheated due to malfunction. Or, to minimize moisture in a tire, to avoid having that moisture freeze at super low temps seen at high altitude. Not really relevant to any cars I know of...
And, yes, racecars do often use nitrogen. Primarily as a means of reducing water vapour, and keeping heat expansion psi of the tire more consistent for handling purposes. And convenience.... just bring a large bottle of nitrogen to the track, and not worry about a compressor, or water vapour etc. Again, not really relevant to most daily drivers...
But, yes, nitrogen molecules are about 3 % larger than oxygen molecules, so tire pressure drops ever so slightly slower... CR found that nitrogen filed tires lost 1.3 psi less on average per YEAR.. yes, per year.
And the other mentioned benefit is less oxidation of the internal tire rubber, promoting longer tire life and safety. Great, except the rubber on the outside of the tire is still exposed to oxygen etc., and will dry out and crack regardless of what's inside the tire... and the tread usually wears out long before anything else matters anyway...
So, if nitrogen is free, great... !