There are many more differences than there are similarities between these two.
The main similarity is that both were conflicts that occurred on a global scale. In both cases, countries around the world were involved and conflict took place in every part of the world.
After that, the similarities end. The Cold War was more of a political struggle for power than it was an all-out war like WWII. There were no actual battles between the two main opponents in the Cold War. Because of this, the smaller countries of the world played a much more important part in the Cold War. In WWII, the two sides were not really competing for the approval and support of various small countries. During the Cold War, the major powers had to try to win the support of small (and newly independent) countries in Africa and Asia.
Though both were world-wide, the Cold War was nothing like WWII because it was a political conflict much more than it was a war.
It is hard for this educator to agree that World War II and the Cold War were about keeping America and American capitalism safe. Years before the Japanese attacked U.S. military bases in Hawaii and the Philippines, Japan had already invaded Korea, Southeast Asia, and China. Germany, under the leadership of Adolf Hitler and the National Socialist Party (the Nazis), had already annexed Austria, seized the Sudeteland from Czechoslavakia, invaded France, invaded Poland, invaded Russia, and threatened to invade Britain, bombing it for years when an actual physical invasion proved impractical. None of that had anything to do with the United States and its form of government or with its economic principles.
World War II was the largest, most destructive conflagration in human history. The Holocaust, the deliberate, systematic annihilation of some ten million people in concentration camps and in mass slaughters of entire villages or towns by German and German-allied troops, represented mankind at its absolute worst.
The Cold War, so-called because a major war between the United States and its North Atlantic allies on one side and the Soviet Union and its satellite nations on the other, never occurred. The Cold War was characterized by two ideologically-distinct blocs of nations facing off across what Winston Churchill, in his speech in Fulton, Missouri in March 1946, called an "iron curtain" that descended from "Stettin in the Baltic to Trieste in the Adriatic."
Had a major conflict occurred between the two sides it almost certainly would have involved the large-scale use of nuclear weapons, with the attendant destruction to both blocs that would have entailed. Fortunately, that conflict never materialized, although the 1962 Cuban Missile Crisis was as close as it would get. Instead of direct face-to-face fighting between the U.S. and the U.S.S.R., the two countries fought each other through proxy forces in Africa and Latin America, with the civil war in Angola and war between Ethiopia and Somalia being among the more protracted and bloody. In those wars, what Rudyard Kipling would have called "the savage wars of peace," either U.S. or Soviet-backed insurgencies waged war against either U.S. or Soviet-backed governments in less-developed countries.
There are many similarities between the Cold War and the Second World War, but there are many more differences.
WWII was a much more "traditional" or "hot" war. That is, countries formally declared war on one another and then fought until one declared victory. In the case of WWII, the victors were the Allies. However, in the Cold War, the U.S. and the Soviet Union never formally declared war upon one another. Instead, they resorted to other tactics, such as nuclear arms races, covert and overt funding of militant groups, the Space Race, and military involvement in many countries' affairs (such as Vietnam and Korea).
The Cold War was more of a war of ideologies and economic systems whereas the Second World War was a war that stopped the expansion of empires.