Why was nuclear weapons important in World War II and the cold war?

I'll answer the World War 2 part. Someone else can improve this answer for the Cold War part.

Actually, there weren't any nuclear weapons used in World War 2. Atomic weapons, yes, nuclear, no. I'm not really sure what difference it makes. Anyway, two bombs were dropped at Nagasaki and Hiroshima at the very end of World War 2 on Japan because the Japanese were essentially making their islands into huge forts and if they tried to invade the islands manually, they projected it would have cost millions of Americans' lives. So the US dropped two bombs. After the first one, Japan was scared, but they thought the US only had one, so they didn't surrender. When they dropped a second, the US said, "Hey Japan! We got a ton of these bombs and we're gonna blow you away!" --even though the US only had 2 at the time. Anyway, Japan surrendered and the Allies officially won World War 2.