The Mongols would have destroyed the Novgorod Republic, which was the nucleus of the rebirth of Russia. There is a good chance that the Middle Volga would remain Finnish speaking rather than Russian speaking, especially if the Mongols went on to conquer Finland from the Swedes.
Interesting question whether the Mongols would have gone on to conquer Sweden and Norway. I suspect that would have depended on whether the Gulf of Bothnia was frozen over around the Vasa narrows in the winter of 1240. If the Gulf of Bothnia was open water and the Mongols had to go around it, probably not, unless the Swedes insulted them terribly. If the Gulf of Bothnia was frozen over, the Mongols would likely conquer Uppsala, then the capital and quickly cross the Kjolen to Nidaros (Trondheim, then the capital of Norway, ending the Olafid Dynasty and perhaps aborting the Christianization of Scandinavia and the Baltic before Ogadei Khan died and they had to return to Mongolia for the Kuriltai to elect the next Khagan.
Peninsular Scandinavia would probably be united early and not unite with Dennmark in the Union of Kalmar. It is entirely possible that the Finns might become the dominant force in Northern Europe, not the Swedes or the Slavs, if the Golden Horde receded. The Finns would likely finally become Catholic unless Tibetan Buddhism came to the Golden Horde in the 14th Century, which would require an additional change, the defeat of the Uzbek Khans in what is now Kazakhstan who converted to Islam and perhaps a reunited Mongol Imperium under the Altan Khans.
A Finnish dominated Scandinavia, stretching all the way to the Atlantic and perhaps with the Livonians, Esthonians and Lithuanians as well as Swedes and Norwegians subject to them would be a major change to the European balance of power since it would also stretch to the Urals, likely with an uneasy Southern frontier with Poland doing the same thing further South Interesting relationships with England and Scotland.

. Eventually, Catholic Finns cross Siberia to the Pacific. Remember, nations build their power because people speaking their language have more children than the people surrounding them.
