For a "fully Axis Finland", an idea that comes up on the forum now and then, you'd need to ask why the Finnish leadership thinks it is a good decision to throw its lot entirely with Nazi Germany, burning all bridges to the Western Allies and making the nation entirely, not just partly, dependent on Berlin's whims?
The only realistic scenario I can see for that is that it is a Fascist Finland, or something very nearly that, with an authoritarian leader that is not Mannerheim. A Fascist Finland, though, would necessarily be weaker than OTL Finland, as the Social Democrats were quite strong in the 1930s and the political left would have to be crushed/subdued internally for Finland to join any German attack on the USSR in the late 30s. This means that this Finland will not be able to mount as effective a campaign as OTL Finland did in the Continuation War, unless with very heavy German aid in the form of troops, weapons and materiel. Remember that IOTL taking part in Barbarossa required the Soviet attack against Finland in the Winter War, in terms of the popular opinion. ITTL, we have a politically divided, partly even rebellious population.
The end result here is that this Finland will not be more help to the Germans in terms of resources, but politically and strategically it might be - in that this Fascist Finland would allow German troops to attack Leningrad via the Karelian Isthmus and might well mean that the city would fall in the early parts of an alt-Barbarossa. There would also be a better chance to cut the Murmansk railway for good. But both of these feats would have to be done mostly with German troops which would mean that there would be comparatively less German troops available for other fronts and theatres.