The weather only allowed for about a month advance of the timetable, but serious issues remain. Yugoslavia would exist even without Greece, but that can be settled to a managable level if Hitler did not insist that they join the Axis. This was not the only problem, but it did turn out to be the straw for the camel's back. So if things do for some fortuitous reason play out and the invasion happens around the 28th of May? Well, there are several issues: the pockets in the Ukraine. Typhoon had to be delayed while Guderian diverted his panzers to close the pocket. This delay cost about a month or more, but the attack on Mosow would not be able to continue anyway, as the rail lines that supported Typhoon were not built yet and really were not ready until after the pockets were dissolved. The extra troops and equipment not wasted in the Balkans would be available, but this would not be enough to close the pockets without Guderian and they would be an extra drain on a taut supply situation. Though a net boon, they are not decisive.
If the Germans do reach Moscow with better weather, the same troops that stopped the Germans' advance OTL are still there and are going to stop them again. The extra time and weather will just help the Germans dig in before the real cold hits. This may actually prevent as many winter casualites as OTL, but it doesn't change the strategic situation. The dilemas of 1942 are still unchanged, but now the Germans have more men and have had some more time to work on the rail system. Butterflies will change the course of the war, but the outcome is likely unchanged.
Barbarossa was a toss of the dice and when Russia does not collapse as expected, the initial "oh fuck what now?" reaction still happens. From here on out, German has lost the initiative and flails about for a feature to capture and win. When the Russian army doesn't stand and fight, they are not captured. Therefore they could not be the goal; one cannot destroy and defeat what does stand and fight. The extra troops not tied down in the Balkans can be used in Russia which means something like 300,000+ troops plus equipment. This also means that the Fallschirmjager can be used, but they are likely to be used against Malta and in African campaign, as there were not many opportunities for their use in Russia outside the light infantry role (I know they were historically used in Russia, but they really were misused and wasted). The partisan situation gets better as the Balkan troops are used in behind the lines ops. That means too that it is likely Russia is destroyed worse, as more men are running around destroying villages and going after guerillas. But it does mean that supply lines are better off, so the massive support that the Russians had from partisans, especially in later war intelligence and sabotage ops., gets reduced by the extra commitment of troops.
It is likely to mean more Russian casualties to move forward and the war takes a bit longer, but nothing major changes in the outcome, just how long Russia survives as a communist state before the mortal wound of WW2 catches up with her.