What would be the differences during WW2 and how would Finlands political landscape look post V.E -day and during the Cold War?
Soviet position is somewhat stronger due to no siege of Leningrad (with continuing industrial production in this industrial center), much better access of Baltic Fleet to Baltic (even if Hanko base is evacuated or neutered via truce or some other arrangement), better access of Northern Route Lend Lease and finally, due to no casualties and demand of forces due to Finnish front. One factor which could also help Soviets might be access to Swedish and Finnish industrial production through trade deals.
As the VE day would arrive (maybe sooner, maybe later, maybe close to historical period, perhaps Soviets would waste their increased capacities in some futile offensive) Finland would definitely and immediately join the Western block at quite early stage, very possibly together with Sweden.
Finnish post-war political, cultural and economic scene would be vastly different. First, due to no significant economic and human damage due to Continuation War (including the reparation payments to Soviet Union), Finnish economy would get a jumpstart incomparable to OTL. More economically healthy Finland I think Finland would be receiver of immigration from Southern Europe rather than a point of emigration to Sweden. Finnish population at 2010 might be somewhere between 6 - 6,5 million.
Also I don't see the powerful SKDL (Communist) party materializing in it's historic force due to no lost Continuation War. Most probably post-war political scene would be more dominated by SDP than OTL.
Culturally, with just one war which would be seen universally righteous, I think post-war Finland would be more militarized than OTL, possibly even more than cold-war Sweden. A joint nuclear weapons program with Sweden might be a realistic possibility. With more affluent and militarized Finland more rooted to pre-war values and culture I think the backlash of 1968 generation would be stronger. This would also pre-empt the romantization of war -phenomena which has been pretty strong in Finland during last 20 years.
And how about art? Väinö Linna would definitely not write Unknown Soldier, although Under the North Star might be even possible hit. Due to no cultural exchange with Soviet Union during 1950's to 1980's many Russian influences to Finnish movies, art, literature, music and design might well not materialize which would be a shame.