I lov that power button and I think(not sure) it just gives you more power and smooth transition+more fuel consumption of course. Once you turn it on, you will see that your car will respond quicker and the engine sound will change slightly (sounds good anyway). Owhen you press that button, it becomes green or yellow, cant remeber exactly what colour it was.
About the second question, it is pretty straight forward, if it is automatic, it simply displays 'D' and if you switch to manual, you will see the numbers. I have never seen S or M. I have 2001 300GT too. About the snow mode, I'm not sure whether you have ever had a chance to drive on snow. Basically, you would not use brake while going down a slope, you would rather use engine brake to slow down the vehicle. Imagine you are driving your car in automatic mode, the car will speed up while going down. Then you need to brake to slow down. This button will possibly do the job for you so you would not need to brake. On the other hand if you brake, the wheels might get locked and then you might loose control. Once again thats what I think.