Though an ardent Nazi, Goering was not a monk-like fanatical party leader like Hitler (or, before him, Lenin in Russia). Goering liked to eat meat, drink alcohol, and use morphine, and enjoyed the good life afforded him as part of Nazi Germany's political elite. However, I suspect that the responsibility of national leadership--and the necessity of projecting a certain heroic, self-sacrificing image during wartime--would have curbed Goering's excesses.
Given Hitler's boldness/recklessness, Goering could not help but be a more cautious wartime leader who actually listened to his military advisors on the German General Staff (at least outside his own area of expertise). It is difficult to see Goering, who emerged as a hero from WWI, opening a second front in the East against the USSR while Great Britain remained in the war as Hitler did. Loss in the Battle of Britain would have humbled Goering, who was leader of the Luftwaffe, at least somewhat. Some sort of armistice with Great Britain might have been possible, perhaps after serious German efforts to starve out the British Isles with unrestricted submarine warfare in 1940-42.
Stalin's USSR would have likely started a war with Nazi Germany by 1943-44. So Goering's Germany and Stalin's USSR would slug it out in Eastern and Central Europe, while the British Empire and USA would remain on the sidelines as spectators in the European theater. The Brits and Americans would be focused 100% on the War in the Pacific against the Japanese after December 7, 1941. The question of whether the Holocaust would have happened under Goering is difficult to speculate upon.