Alex,
According to the science (Avogadro's Law-you can look it up), the tires should be the same when cold, if that's when you fill them. When heated to 100 Celcius, (possible only on the racetrack) the variable will be less than four-tenths of one PSI (.4psi).
This assumes: Your Nitrogen tires were filled with 100% Nitrogen. This is impossible because 100% pure laboratory grade Nitrogen would never be used to fill tires and the process of filling tires with Nitrogen never eliminates all the compressed air (you would need a vacuum chamber).
This also assumes: That the compressed air tires were filled when there was 100% relative humidity and no drying device was used in the compressed air process (unlikely).
If your tires were filled with 100% compressed OXYGEN at 100% relative humidity, the maximum possible variable of the partial pressure of the water vapor would be about 2 psi. However, compressed air is ALREADY 78% nitrogen so the variable is much less.
I regularly read, with disdain, the marketing literature of the Nitrogen industry. They promise better fuel economy, better handling, longer tire life, reduced carbon emissions, save the whales, etc, etc, etc.
There is just no science to support these claims. Sorry.
A good tire gauge is all you need.

HOWEVER, the placebo effect of Nitrogen is well documented in these forums and many others. Certainly, Nitrogen won't hurt your tires. Use it if you wish.
But the money spent would be better used on your favorite charity or a nice bottle of Chardonnay.

 

Yes I recall good old PV = nRT way back from school.

£6.80 ($10) would get you a glass of Chardonnay not a bottle!!! LOL. It's a no-brainer at that price.

I have compared my cars back to back with air and N2 and there IS a difference according to my TPMS readouts.