This is a good question with little firm answers. The one man who knows is Stalin, and he is dead.
However, historians believe that among the factors were
Stalin had actually quietly dropped his support for the puppet government after January 1940, as the Red Army wasn't able to crush the Finnish resistance as planned. However, there are reasons to believe that he intended the 1940 Moscow Peace Treaty - that emasculated Finnish defensive lines - to be only an interim peace, followed by a renewed attack once the threat of Western intervention had passed. Accordingly, the Soviets firmly refused to permit Finland and Sweden to join into a mutual defensive pact, even though as one of the preconditions the Finns had officially renounced any claims to lost territories.
Furthermore, during summer 1940, Soviet military buildup against Finnish border was alarming enough so that Finland came very close to ordering full-scale mobilization. It is also known from captured German archives that during that time, Soviet foreign minister Molotov asked the Germans for a "free hand" to move against Finland and finally complete the 1939 Molotov-Ribbentrop agreement's agreed spheres of influence. At this time, however, Hitler refused. Planning for Operation Barbarossa was by then well underway, and Hitler saw Finland as a possible ally.
The Finnish army was certainly on its last legs by March, but it was still fighting hard - it had not collapsed and it had not been crushed. While Mannerheim told the government that further resistance was only possible with immediate direct intervention of foreign regular forces (meaning Sweden, as it was the only friendly country close enough to provide troops immediately) or by sacrificing the army to allow the population and government to flee to Åland and perhaps Sweden, the Soviets did not know this at the time.
The Western Allies were also making serious and public attempts at supporting Finland militarily. The French were drawing up plans to bomb the Soviet oil fields at Baku from bases in Syria and the British and French were forming a corps and requesting the right to transit it to Finland from Sweden and Norway, intending to land it in Narvik and go by the ore railroad to Luleå and from there into Finland.
Stalin was a brutal man, but he was also a careful opportunist in his foreign policies. He viewed the Molotov-Ribbentrop pact as the perfect tool for him to deal with Eastern Europe (and Finland) while the Western Allies and Germans fought each other to a stand-still on the Western Front.
In March 1940, it started to look like his plan was failing - the Western Allies and Germany sat on their arses watching each other in the Phony War, while the Western Allies were gearing up to support Finland directly. The world's public opinion was also turning hard on the Soviets, and Stalin probably feared that the Germans and Western Allies might actually make peace to fight him instead - quite the opposite to his intention with the Molotov-Ribbentrop pact. The fact that Sweden had an army corps mobilized and ready at the Swedo-Finnish border and that international and Swedish aid was flowing into Finland would most likely also factor into his decision on the matter. The Soviet minister (a diplomatic rank lower than ambassadors, as before 1945 only grand powers had ambassadors in each other, minor countries and to minor countries, you most often had a minister heading a legation rather than an ambassador heading an embassy) in Stockholm, Mme Kollontaj, warned that the Finns had officially requested direct Swedish military support, and that the Swedes were preparing to send two fully equipped divisions as volunteers. This plan was abandoned due to the German military attaché in Sweden threatening Sweden with war over the issue, but the Soviets did not know that at the time.
The Swedes were working hard diplomatically to achieve a peace, as the Western Allies were making threats that they would land in Narvik and go to Finland over the ore railroad with or without Norway and Sweden's consent, while the Germans informed Sweden that any Western Allies troops on Swedish soil would force the German to intervene. Since the Soviets were smarting from Swedish aid to the Finns during the war, Sweden faced potential hostile action from all three Grand Power blocs of the time - not a very rosy prospective, to say the least!
The bottom line is that the Terijoki government was not worth the risk of a Western Allied intervention in the mind of Stalin. The Soviets thought they could deal with Finland later, when the Germans and Western Allies were pre-occupied with each other. They also tried, blocking Swedo-Finnish attempts at forming a state union in early 1940, conducting multiple provocations over the new border in Summer 1940 and entering talks with Germany about "solving the Finnish issue". However, by Autumn 1940, the Germans saw the Finns as potential allies in the upcoming conflict with the Soviets, and were selling captured arms at cut-rate prices to the Finns, and stationing troops to protect the nickel mines at Petsamo, finally giving the Finns the military aid they desperately sought.
To summarize:
The Finnish army was on its last legs, but was not crushed and still fighting hard. The Soviets did not know how desperate the situation was for the Finnish army when they made peace.
The Western Allies were preparing to intervene directly. They intended more to occupy the Swedish iron mines at Kiruna than aiding Finland, and both Norway and Sweden refused them transit but Stalin did not know this.
World opinion was decidedly against the Soviets, and aid was streaming to Finland, especially from Sweden. What had been intended as a swift occupation and take-over had turned into an embarrasing slog that was seriously harming Soviet international good-will.
A direct Swedish intervention, either by sending troops as volunteers or simply allying with Finland and entering the war seemed probable to the Soviets at the time, aided by information passed on by Mme Kollontaj in Stockholm.
Despite Germany doing its best to abide by the Molotov-Ribbentrop pact and stopping among others Italian and Hungarian shipments of arms, supplies and volunteers over German railroads and threatening Sweden over its aid to Finland, the Soviets did not feel safe that Germany would maintain this attitude. The lack of fighting between the Germans and the Western Allies on the Western Front seemed to indicate this to the Soviets.
The Soviets believed that they could "take a break" and deal with Finland again when the international situation was more in their favour (ie the Western Allies and Germany fighting each other and the world being focused on that).
With this in mind, the peace seem like a logical option for the Soviets.
Sources:
Kriget kommer! Den hemliga svenska krigsplanläggningen under vinterkriget 1939-1940 (The war comes! The secret Swedish war planning during the Winter War 1939-1940) by Arvid Croneberg
Liten kugge i stort spel - en stabsofficer berättar (Small cog in a big game - a staff officer speaks) by Leo Wikman.
Sveriges militära beredskap (Sweden's military readiness), collected reports by the British and German Military Attachés to Sweden, Sir Victor Mallet and Bruno von Uthmann 1939-1945.
Andra världskriget och Sverige (The Second World War and Sweden) by Jan Linder.
Svenska kryptobedrifter (Swedish decryption deeds) by Bengt Beckman.
Ofredens hav, Östersjön 1939-1992 (The Sea of War, the Baltic Sea 1939-1992) by Jan Linder and Lennart Lundberg.
Med döden i hälarna, högkvartets fjärrpatruller 1939-1945 (With death on their heels, the headquarter's long range patrols 1939-1945) by Erik Appel.
That kind of happened twice, first in 1940 after the Winter War and second in 1944 after the Continuation War.
In 1940 the Winter War had turned out to be a bit of a humiliation to the Soviets. Finland had been able to resist the invasion for much longer than the Soviets had thought possible. Secondly, there was a risk of intervention by the western allies, primarily the UK and France. This was during the 'sitzkrieg' or Phony War period on the Western front, before Germany had occupied France. Both Western powers still considered the Soviet Union a great threat. The Finnish government was conducting negotiations with the UK and France about direct military aid. The UK had promised that aid to Finland would arrive in April 1940, which prompted the Soviets to offer peace to Finland in which Finland would retain its independence but would cede large amounts of territory to the Soviet Union, including a sizeable buffer zone around Leningrad, which had been the Soviet Union's primary war goal. There was still a faction within the Finnish government that favoured continuing the war while waiting for Western assistance despite the severe depletion of the Finnish military. Uncertainty about the number of Western troops and the timetable of their arrival delayed Finland's response to the 1st of March 1940 deadline set by the Soviets. That uncertainty eventually caused Finland's government to want to negotiate on 6th of March, which the Soviet Union accepted despite the original deadline having been passed.
In 1944 the situation was very different. Finland was de facto allied to Germany even though no formal alliance had been established. Finland was a fairly small and unimportant part of the larger war so the Soviet Union wanted to end that part to be able to fully concentrate on Germany. This was especially important after the Normandy landings and subsequent "race to Berlin". To deal with Finland, the Soviet Union launched a major offensive on 9th of June 1944. Finland was nearly defeated but president Ryti made a secret deal with Germany, in which he acquired German material aid in exchange for a promise to stay in the war to the bitter end. That was enough to practically stop the Soviet advance in the end of July 1944, for the moment. Enough to make the Soviets evaluate their larger war aims and consider peace with Finland. Another issue was that communism was still a force within Finland despite having been heavily suppressed after the Civil War, especially in the 1930s. The Soviets thought that there was a chance of a communist takeover after the peace was made. In any case, the territorial cessations made sure that there was a good enough buffer zone around Leningrad and Finland was to pay sizeable war reparations. All that made peace an attractive choice for the Soviets since they would be able to shift all those troops and equipment South to fight against Germany afterwards.
To summarize:
In 1940
Soviet primary war goal achieved.
Threat of Western assistance to Finland and ending up in a war against the UK and France.
In 1944
Finland managed to (temporarily) stop the summer offensive of 1944.
Race to Berlin and the Soviet wish to focus on Germany.
A chance of creating a friendly government in Finland anyway.
Source:
Suomen historian pikkujättiläinen, 2003
ISBN 951-0-27365-1