Goring had the force of personality and the charisma to maintain his power,something nobody else in the Nazi leadership had. Even tt the Nuremberg Trials, the other Nazis deferred to him and commentators spoke of the dominance he exerted over most of his co-defendants.
The Holocaust would not have happened, since only Hitler had the level of vitriol to carry anti-semitism to such an extremity, but the Jews would certainly have still had an unpleasant time.
Goring was opposed to Operation Barbarossa, so it is possible there may have been more alignment with the Soviet Union, or at least a benign neutrality. Goring was quite indolent, so would have been quite happy to leave the conduct of land operations in the hands of the professionals. This may have had a detrimental effect on the attack in France, since it was Hitler who backed Manstein's plan, and as a consequence the Western Front may have degenerated into stalemate or at least a more costly and protracted campaign. Goring, however, would have been satisfied with territorial gains to date, and may have been prepared to restore a rump Polish state in exchange for peace. Maybe the Phoney War would have been indefinitely extended, resulting in a peace through Goring being prepared to reach such a compromise, and more generally through inertia. In the long run. as the inherent contradictions and absurdities of Nazi doctrines became apparent, Germany would probably have reverted back to something like the Kaiserreich.