One would certainly THINK there would be no differences, but the Finns generally have a pretty good idea about what they can get out of a rifle... and they are still using old-fashioned octagonal receivers on their sniping rifles. It certainly is not because they haven't got a new rifle since 1917, either: Russia has lost a pile of rifles every time they have gone into Finland.... and the round-receiver rifles were the ones the Finns sold off first.
When you start talking about real precision in rifles, you want the most solid action you can get your hands on. This is why top target rifles on Mauser actions sometimes have the charging-slot welded solid again, to give a stiffer left-hand receiver rail. The heavy, stiff old octagonal receivers of the early Moisin-Nagants could well be performing a similar function.
The way to find out would be to build two absolutely identical rifles with differing receivers, then shoot them until they are worn out, keeping track of every single shot along the way. I certainly have never done this, but I do rather think that the Finns might have. They must have SOME reason for their preference of the old-style receiver and it can't all be esthetics: when the chips are down, the rifle which is the best is the one that kills your enemies, not the prettiest one. There are no people in the world who know this better than the Finns.
"Ah, the Romans were wusses, Sid," said Aimo, "They never came up against the Finns. Only the Finns can rule the Finns; nobody else ever lives long enough." -- Aimo the Wizard in "The Little Red Hawk".