There are several things to consider when you try to choose the best time for traveling to Russia:
        Location
        Budget
        Purpose of travel
        Health condition

Location

       The location is important because many big Russian cities are situated in the Northern part of Russia which makes them quite less appealing in winter. It is not just about severe Russian winters, but rather about fewer daylight hours. If you go to Saint Petersburg in winter, you'll see that the sun goes down at approximately 4 pm which leaves little time for exploration and even visiting museums which may close earlier in winter time. If you travel to Saint Petersburg in summer, it will be sunny till up to 11-12pm, and in June you are likely to witness famous white nights.

Budget

       You may save up some money while traveling to Russia in winter. Airline tickets and hotel rooms cost less during winter months due to the lack of tourists. However, don't forget that saving on travel expenses may turn into saving on good impressions about this country.

Purpose

       Depending on where you are traveling to Russia in winter, you can engage in skiing, skating, hiking or mountain climbing. However, traveling around and seeing different cities and Russian countryside are more pleasant in summer. If you travel to Saint Petersburg, you can see white nights in June.

Health Condition

       It is hard to live in some cities near a body of water in summer. A lot of vapor makes the summer air very humid. Such cities are usually on the Volga River and in the south of Russia near the seas. When the weather is warm, the humidity makes it hard to breathe and deal with the heat. That is why if you cannot bare hot weather very well, choose late spring or early fall for traveling to Russia.

       Also if you have a serious medical condition which may require urgent medical help, do not go to any cities other than Moscow and Saint Petersburg. There are few clinics for foreigners in Russia, and they are all in these two cities. Russian clinics and hospitals may not even try to treat you, as you are not a Russian citizen and do not have Russian medical insurance.