Why does it actually get warmer when it snows?
Snowing does not cause the warmth, the relative warmth causes the snow.
During very cold weather the capacity of the air to contain moisture is greatly reduced. What water vapor remains is deposited as frost. At the same time, evaporation is reduced and therefore air humidity remains low. In these conditions snow cannot form.
Warmer air can hold more moisture before saturation point is reached and the water vapor begins to condense out. So when warmer air moves into cooler areas, not only do we feel the increase in temperature, but the excess vapor precipitates as snow.