Not much of a difference really. Same cold war but with the Third Reich instead of the USSR. Same same.
Germany already declared war on the USA in 1941. If they had assimilated the power of the USSR into their own (and frankly, Stalin’s ideology was more national socialist than international socialist), it would have been a hot war.
Nazism did not contain within it the seeds of its own destruction like communism did. They would have been a stronger and thus worse enemy than the Commies were.