Dry nitrogen will contract and expand less under temperature change. Normal air (~78% nitrogen, ~21% oxygen, ~1% other) with humidity in it will change tire PSI by about 4-6PSI between a cold tire and a tire at operating temp. A dry nitrogen filled tire will only increase pressures by 1-2PSI. As the seasons change an air filled tire will drop about 1PSI per 10 degrees (F). Dry nitrogen is less ( I don't remember how much less). Dry nitrogen is also slower to seep out of the tire.
It has to be DRY nitrogen. Any small quantity of water vapor in the nitrogen almost completely negates the benefit.
Nitrogen filing makes sense if you are auto racing and 1-2PSI makes a difference.
Me... I use air. Good old fashioned air! Filling your tires with nitrogen is pure MARKETING WANK! There are legit benefits but they are minuscule, i.e. ~0.5% fuel economy improvement... but only during the period before your tires warm up. Also, vehicle manufactures designate tire pressures knowing they will warm up and pressures will go up, so if you fill your tires with nitrogen to the manufacturers specification, you will be driving on perpetually under-inflated tires...
Nitrogen filled tires are wank. I don't judge anyone that chooses to do it but... if you've paid $500 for it!?
That dealer needs to be flogged.