The best time is throughout the year except the monsoons. For a good time, go in the summer but if you want to enjoy the snow, go in the winters. Both the time is good, so choose you're best time according to your choice. Happy travelling.
Unpredictable is the first word that comes to my mind when I think about describing the weather in Gangtok. December, January, February is extremely cold with March, April, September, October, November being pleasant and cool. May, June, July, August are the catchy months as daytime will be hotly followed by heavy rain in the evening, landslides causing road blockage is a prominent phenomenon during these months. Do pack thermals, heavy jackets along with muffler, caps and a lot of socks if you plan to come in the winter time. If you plan to visit in any other season do carry light sweaters and an umbrella. Also, moisturizer and lip balm is a must carry.