There are many ways it would have an effect on WWII, starting with the fate of Norway. Given the plans about Allied intervention in Finland, the Winter War continuing after mid-March 1940 could very well mean that the Allies move sooner on Norway, and thus the German conquest of Norway is avoided. Most parts of Operation Pike would probably be avoided, though, as by late March it would become obvious that the Allies will never reach Finland in time to stop the Red Army take Helsinki.
The obvious changes to the OTL would include the position of both Norway and Sweden. With the Allies in Norway and the Red Army in Finland, the Swedish would have to take the Allies' views into account much more than Germany's ITTL. The result would be less Swedish goods for Germany. Sweden would also receive a lot of Finnish refugees, which would be a major political issue for the Swedes, as would be the presence of Soviet units near to its borders. For the Soviets, control over Finland would be a definite boon. TTL would probably avoid a Siege of Leningrad, and most likely the Murmansk railway would not be under real threat. With Norway in Allied hands, LL shipments to Murmansk would also be safe. The occupation of Finland would require some Soviet troops, but when the various losses the USSR suffered due to the Continuation War would be avoided, the comparative benefits would be significant. All the resources Finland sold to the Germans would now be controlled by the Soviets - including Petsamo nickel.
Now, it is difficult to say whether in TTL's *Barbarossa, assuming it is launched in a similar manner as IOTL (and if the Battle of France panned out pretty much as IOTL), the Germans would attempt landings in Finland. Given that they would likely have more naval assets in the Baltic Sea than IOTL, after taking the Baltic states a plan to invade Finland would at least be floated, to open a new front against Leningrad. German control of the Ålands and a foothold in Finland would also be good for pressuring Sweden and protecting iron ore transports from Luleå. How would a German invasion of Finland across the Gulf of Finland work out? Even if the Germans manage to land in Finland in the fall of 1941 or in the summer of 1942, their position would still be weaker in the northern theatre than IOTL. They would lack the manpower and all other assets of an independent Finland here, even if they would have a number of anti-Soviet/pro-German "Free Finnish" auxiliaries to support them.
Most likely TTL's war ends sooner, with the Soviets beating the Nazis more easily than IOTL and advancing further into central Europe than in our world. Millions of less people would die in Eastern Europe, and the Holocaust would claim less lives as well. In the end the *Cold War would start with a stronger USSR in a better position than IOTL. The one clear loser ITTL would be Finland, which would lose at least hundreds of thousands more people (as casualties of war, as refugees and due to Stalinist repression), even more if there is a successful German invasion of Finland, one that will mess Finland up thoroughly before the Germans are defeated.
The Finnish SSR would most likely include OTL Finland and most of the area of the OTL Karelo-Finnish SSR. By the 1980s, it would probably have a c. 30% Russian minority, just like the OTL Baltic states did. Post-Soviet Finland would in many ways look pretty similar to the OTL Baltic states as well, especially Estonia, though ITTL it is hard to know when (and if) this USSR falls.
As for Sweden's membership in NATO, I'm personally of two minds about that. Depending on different developments during and after the war, both Sweden joining *NATO (as long as there is a recognizable NATO ITTL) and becoming a more or less "Finlandized" neutral are possible, IMO. Of all the possibilities, a *Warsaw Pact Sweden is the least likely option.