Ironically, the German invasion of Russia was made into a muck by the earlier Italian invasion of Greece. The Italians got their asses kicked around by Greek forces, and Germany had to pause planning for BARBAROSSA until they'd reinforced Italy's position. The delay proved fatal: invading in the middle of the summer (crossing the border on the same day of the year that Napoleon did. Woops) meant they wouldn't be able to knock the Soviet Union out of the fight before General Winter came to her rescue.
The Germans invading sooner just means they are beaten at the gates of Moscow earlier in the year. The cold certainly didn't help, but what killed Army Group Center was the overexertion of it's logistics at a time when it was facing ten Soviet field armies outside Moscow.
Actually, Germany could not have invaded earlier than they had.
Most people tend to think that it was "General Winter" that defeated the German Army, that is wrong. What actually foiled them was Raputitsa. The "Season of Mud".
Which is actually somewhat of a misnomer, it is actually "seasons".
There are 2 time periods where invasions of the Soviet Union was not possible. Spring and Fall. In the Fall you have the Fall Raputitsa, this is what ultimately ended the German advance on Moscow. It generally starts in mid-late October, and makes almost all road travel impossible. And this season ends when the ground freezes in early November.
Then you have the Spring Raputitsa. This is the worst one, because not only does all the mud that had been frozen thaw out, but it is compounded by the winter snows and spring rains. And the German Army had both of these seasons work against them in 1941.
First, the Spring 1941 Raputitsa was longer than usual, lasting until early May. There is simply no way the German Army could have invaded sooner than that, they would have become quickly mired in the endless mud flats of the Western Soviet Union if they had tried. They had originally planned to invade in May (normally the mud is gone by then), but in 1941 they could not do this.
And their entire battle plan was to rush for Moscow and secure it before the fall Raputitsa set in. Or failing that, to hold in place until the winter freeze allowed them to move again.
But things worked against them in 1941. The rainy season that year started in early October, and the freeze did not come until early December. That was almost 2 months of little offensive actions, and few supplies reaching the front lines. By the time the ground was hard enough to resume the campaign, the Soviets were ready and we all know what happened then.
This was their own territory, and they knew what to expect. The Soviet Army had equipment that had evolved over centuries for dealing with those conditions. The Germans had none of that and suffered for it.
