Mongolia's winter is harsher. Russia back then was centered around Kiev, quite a bit more southern than Moscow, it's only when the Mongols wrecked the region that the Moscow Duchy gained prominence.

Anyway look at the modern climates for Kiev, Moscow and the modern capital of Mongolia Ulaanbataar. Even after taking things like climate change over the last 8 centuries and the fact that Mongolia lost a lot of southern land (Inner Mongolia) to China into account, it should give you a basic idea of the kind of cold Mongols would have been used to.

Russian winter was like spring time to them, frozen rivers were used as highways and Mongol horses are quite special and hardy. In winter back home they fend for themselves, they dig under the snow to graze and can survive ridiculously low temperatures, so they wouldn't have had as much trouble feeding themselves (and Napoleon's campaign was cursed with a drought which made the lack of grazing even worse).