AFAIK the Soviets' interest in Finland revolved almost entirely around one single issue - the defense of Leningrad, one of their more important cities and main Baltic port. One look at the map of the Baltic should be sufficient to illustrate why they had a keen interest in both the northern and southern shore of the Gulf of Finland concerning this topic. (The Finnish border in Karelia being damn near within artillery range of the metropolis didn't exactly put the Soviet strategists at ease either.)

Hence why the terms that ended the Winter War included a Soviet base in Hankoniemi - other than putting potentially useful pressure on the Finnish capital Helsinki (it's about half an hour away by train), in combination with the Soviet grab of the Baltic states emplacing heavy coastal batteries there allowed them to dominate the sea approaches. The postwar Porkkala base performed a similar function; at least part of the reason the Soviets gave it back was technological developements which rendered it unnecessary, as the southern coast alone was now sufficient to control the Gulf if need be.

I've been given to understand that during WW2 Churchill - acting perhaps partly on the "fennophile" sentiments he'd developed during the Winter War, and certainly on the longstanding English strategic interest in the Baltic - cut a deal with Stalin to leave Finland independent, although that certainly didn't keep Uncle Joe for trying very hard to grab as much real estate and advantageous negotiating position as the Red Army could get him. Finland was much too small a fry to piss the Brits (and by extension, the Americans) off over, so as long as the Soviets got the security they sought for Leningrad they didn't have too much against the arrangement.
They had way more important "strategic depth" territory to grab down in Europe proper, anyway.

Post war, Moscow also flat out found it convenient and useful to have a "friendly neutral" minor state - whose politics they had easy enough time influencing - next door as a trading partner and Baltic buffer.