Does leaving snow on my hot tub cover save me money? Snow is supposedly such a good insulator, that some animals dig snow caves in which to hibernate through the winter. However, my hands seem to feel more comfortable being exposed to winter air than to snow, but I'd imagine part of this has to do with the increased heat transfer associated with the water the snow melts into. 
Will my outdoor hot tub run more efficiently with snow on top? Could the answer change depending on air temperature and/or wind speed?
 A: 
However, my hands seem to feel more comfortable being exposed to winter air than to snow

Air (still air) is a better insulator than snow. In fact the insulating properties of snow are due to the fact is has a lot of trapped air within it. The other component of snow, water, is not noted for being a good insulator.
The trouble with using air as an insulator is that it's hard to keep the air still. Even if there is no wind the convection currents caused by your hands/whatever/tub will cause the air to move. And as the air flows away it carries away heat and brings fresh cold air into contact with the object. Though your hands may feel warmer in still air then in snow, I bet they'd feel warmer in snow than in a 100mph blizzard.
If you look at how animals use snow burrows they typically rely on their fur (or blubber if they're seals) as the primary insulator. The temperature immediately outside them will typically be minus a few degrees C. Where the snow comes in is by insulating them from the -40°C gale that's howling outside their burrow.
In the case of your hot tub, assuming the polystyrene lid you mention can provide enough insulation that the temperature above it stays below 0°C then a layer of snow would add more insulation if the external temperature is well below zero. If the external temperature is only around zero then the layer of snow will achieve little.
Snow if obviously a useless insulator if the internal temperature is above zero. In that case the snow will melt and the latent heat of fusion required will cause a massive heat loss.
A: Yes, but probably not much.
Snow is great as far as natural insulators go, but your hot tub cover is probably a much better insulator. Adding extra insulator on top won't hurt, but the top of the hot tub cover is already probably quite close to ambient air temperature so heat isn't being lost too quickly by convection.
If much heat is being lost through the sides or bottom, it may not help much, but if not it could be significant.
