Stalin overestimated the efficacy of the Finnish Communist Party and underestimated the canniness of Finnish politicians. Starting in leverage high grade military equipment from the Germans which allowed the Finnish forces to stage a fighting retreat from Karelia in 1944. Thus in mid-1944, the Finns and the Soviets were fighting in the same ground as the Winter War. Both the Kremlin and the Red Army's leadership were much more interested in maintaining the drive into Eastern Europe than refighting what had been a dark chapter in Soviet military history.

Urho Kekkonen, a Finnish parliamentarian and later Prime Minister, said in a 1944 radio broadcast "the Soviet Union must stand to gain a bigger advantage from an independent Finland clinging to life than from a broken Finland doomed to a dependent existence." The cornerstone of Soviet-Finnish relations was the Agreement of Friendship, Cooperation, and Mutual Assistance signed with the USSR in April 1948. The Treaty guaranteed that Finland would aid the Soviets against "Germany or its allies" and fostered a series of networks and political connections between the Soviets and the Finns. The Soviets initially expected the Finnish Communist Party (SKP) to make electoral gains, but the existing Finnish political establishment adroitly managed to sideline them. The Treaty and the Finnish compliance with it did not give the SKP any major issues with which to attack the existing governments. Successive Soviet governments wanted the Treaty to be expanded and pull the Finns closer into the orbit of the Soviet sphere, but the Finns were able to strategically drag their feet. For example, the language "Germany or its allies" meant that Finns were able to justify not wanting to take defense steps against NATO Norway and Denmark. At the same time, the Finns also mastered the art of not appearing to be undermining the larger issue of Soviet security; they would give way over key debates like radar stations its early warning network.

The success of the Finns looks quite intelligent and unexpected from the vantage point of 2014, it's important to keep in mind that during the Cold War the West was quite apprehensive the Finnish policies of accommodation. "Finlandization" became a pejorative term within Western Cold War discourse and a shorthand for making concessions to gain at best temporary freedoms from the USSR.

Sources

Jakobson, Max. Finnish Neutrality; A Study of Finnish Foreign Policy Since the Second World War. New York: Praeger, 1969.

Jussila, Osmo, Seppo Hentilä, and Jukka Nevakivi. From Grand Duchy to a Modern State: a Political History of Finland since 1809. London: Hurst & Company, 1999.

Luostarinen, Heikki. "Finnish Russophobia: The story of an enemy image." Journal of Peace Research 26, no. 2 (1989): 123-137.

Rentola, Kimmo. "From half-adversary to half-ally: Finland in Soviet policy, 1953-58." Cold War History 1, no. 1 (2000): 75-102.

The aims of the Soviet Union had changed as the war progressed, where real expansion into central Europe became viable to the soviet high command. At that point Finland had changed from desirable target for easy territorial expansion to a distraction from the drive to Berlin.

Finland's performance in the Winter War also pointed to a difficult conquest, even with the vast improvements in soviet military technology, training, and leadership.

A separate peace with Finland was also acceptable, as Finland had never integrated it army into the Werhmacht command structure or signed the Tripartite agreement. The western allies were somewhat sympathetic to Finland as well, and never had engaged in combat directly with Finnish troops -- allowing the Soviets to pursue a separate peace.

Finally, Finland made numerous concessions as well, including the forced expulsion of German troops from Finland, ultimately concluding with the Lapland war.

Stalin didn't so much allow Finland to escape communism as fail to bring the Communist Party of Finland (SKP) to power.

Remember that Marxists-Leninists in East Germany, Poland, Hungary, Romania, and Bulgaria all came to power while these areas were under Soviet occupation^1 . Only in 3 places in Europe were they able to take over without the Soviets there to help directly: Yugoslavia and Albania, where they had been part of the resistance, and Czechoslovakia, where they were popular enough and their rivals reckless enough that they were able to position themselves so as to launch a coup. Finland was not occupied by the Soviet Union, it had not had any serious resistance^2 and the SKP, though it won almost 1/4 of the votes in 1945 as part of a coalition, was not able to maneuver its way to power the way the Communist Party of Czechoslovakia did in 1948.

^1 The Red Army withdrew from Bulgaria in 1947, a year after its communization.

^2 I'm not counting the Lapland War, fought against German forces in Finland in 1944-45, as it was waged by the same Finnish government that had previously been allied with Germany.

Would it have been related to the famous napkin he split Europe with? Was it a part of the deal he had with the west that Finland was in the western sphere and would not go communist?

Actually, come to think of it, how corroborated is the story of the napkin? Anyone know what I'm referring to?

Was Finland part of the Allies or Axis? I always thought they were kinda their own little thing being for themselves against everybody.

It was asked from Stalin and Molotov. Not sure were they telling the truth:

According to Tito's own words, he asked this in 1945 from Stalin. Stalin answered: "We took Americans into consideration too much, but they wouldn't have lift a finger".

In 1974 Molotov answered the same question: "We were wise when we didn't annex Finland to the Soviet Union. It would have been an open wound. Not Finland by herself, but the others, they defended her so much - And the people there. They are stubborn, really stubborn. It would have been a dangerous minority".

The soviets were fully committed to fighting the Germans and allocated all resources to that. After which they were depleted and couldn't expand their war effort to try and take Finland again. Also, the soviets had taken a pretty bad defeat to the fins in the 39-40 winter war and beyond maintaining a secure border was not interested in trying again. Finland was the only country in ww2 that managed to play both sides of the war and win without getting invaded, they sided with the Nazis early on, then when the war turned against the Nazis they sided with the allies and met some conditions that kept the allies out.