I like running a lot but also skiing when the winter season hits. With both sports you get the same endorphins, a change of scenery, and some time outside. With running you get your calves and lower legs used more. With skiing however, you get your quads working extra hard! Skiing is for sure a good alternative for those cold temperatures especially if dressed warm enough and ready to work hard with those different muscles which can really help your chances of improving better at running the following summer.
Skiing helps those legs bend more throughout the duration of movement and your need to balance. Being able to balance well on skis when gliding downhill or along a straight forward path works the core really well. In addition the arms and shoulders get a workout if using your poles for propulsion. However, poles are only a substitute towards maintaing your balance. The key to properly skate ski is to skate without poles by pushing forward your whole body and gliding from side to side.
Cross-country skiing is a great full body workout and a great alternative to running!