Lightning and thunder during a snow storm is uncommon. As far as I know, more uncommon than during a typical rain storm. Why is this? I speculate it might be one, or both, of the following two ideas, one having to do with a change in the dielectric, the other having to do with a change in the catalyst.
I can't point to a definitive reference, but my recollection is that thunderstorms are associated with a lower layer of warm air rising rapidly through an upper layer of cold air. It's the rapid vertical transport that generates the static charge and hence the lightening. In winter it's rare to get these atmospheric conditions.
So it's not that there's something special about snow that stops lightening, it's that the atmospheric conditions in very cold weather aren't conducive (no pun intended :-) to lightening.
The mechanism by which lightning is produced is complex and imperfectly understood, but we know moisture is important in two respects:
Heat is released when water vapor in the air condenses into liquid drops, and this heat helps provide energy to the thunderstorm.
Interactions between supercooled liquid water droplets and ice crystals in the upper atmosphere (about 15,000 to 25,000 feet above sea level) are needed to generate the electrical charges that accumulate until a lightning bolt occurs.
Cold winter air typically doesn't contain a lot of moisture and so isn't conducive to thunderstorms.
Air instability is also important to thunderstorm formation. It commonly occurs when warm air near the surface of the Earth rises due to convection on a sunny summer day when the ground gets hot and warms up the air immediately above it. As the hot air rises, cooler air descends to replace it. If conditions are right, strong updrafts can form that quickly move the warm, humid surface air up to the higher reaches of the troposphere, where the water vapor in the air cools and condenses to fall as rain (or ice, if it's cold enough). These updrafts are a hallmark of thunderstorms--the strong upward motion of the air encourages the interactions between water droplets and ice crystals that can lead to lightning.
In winter, cold surface air temperatures and reduced sunlight mean there's less surface heating, less convection, and thus fewer opportunities for thunderstorms.
It is simple:-
Snow is solid distilled water (high resistance and dielectric break-down).
Rain water is also distilled water but during its journey to ground, some acidic gases like ( NO2, SO2, SO3 ... etc. ) mix in the rain water droplet and hence becomes better conductor than snow.
I doubt if this question is a Physics question.
Expounding upon John Rennie's answer since this question is still open. From a thermodynamic point of view: All of the constituents involved in lightning are terrestrial, and we haven't observed anything to have a negative heat capacity naturally so far. We know that if we model a region where a lightning strike forms and travels some, we can say that this approximate region itself has a specific heat capacity $ c_0 $, that is the average SHC for the region. Lightning merits energy transfer, governed by $ \Delta Q = mc\Delta T $.
But let's say that we spike down the temperature of our system to bone chilling temperature. $\Delta T$ is going to be negative, and depending on how far down we spike the temperature, very negative. This will correspond to a change in energy of our system. We know at the very least, that a lightning strike again, is a reaction of energy transfer. At this point, there will be significantly lower amount of energy in the surroundings to even generate a lightning strike ( since all the particles are moving around slower as a result of having less thermal energy ). If a strike were to still occur, in it's travel path, there are yet still regions ( ie in Winter ) that have low thermal energy content. Entropy requires that energy flow from concentrated useful forms, to unordered, less useful forms: Heat! The lightning strike will have to pay a hefty energy toll for the air in the travel path to move, allowing it's wrath, but with a low enough temp or large enough distance, this will either completely disallow a strike, or make it relatively small to the monsters we hear in summer storms.
In the end, it all comes down to moving particles, as the other two responders have stated. Particles with energy close to rest energy are going to be harder to move than those that already have some thermal energy: Something the cold saps out.