how did the creation of the nuclear bomb impact American society?

Answer 1

First, it made possible a quick end to WWII. This saved many thousands of lives, both American and Japanese, by cutting the war short. At that time, the public had no previous knowledge of this weapon, so everybody was amazed.
Later, the implications of such a weapon began to be apparent. First the effects of radioactive fallout, short term and long term, became known. And it wasn't long before our latest enemy, the Soviet Union, also had the Bomb, and plenty of it. This created a situation of fear on both sides. For the first time in history, human beings had the capability to end all human life. This led to a strong anti-nuclear movement as well as cooperation between enemies to prevent the next nuclear war. In WWII it was one-sided; now there was the prospect of "mutually assured destruction" ("MAD"). A huge amount of money was spent on nuclear weapons and delivery systems, with the objective of deterring the Soviet Union from attacking us. Eventually the Soviet Union couldn't keep up the pace of the arms race, and so it broke up into component states.
The Boomer generation grew up with the possibility of nuclear war, and this produced considerable angst.
Somehow, the demise of the Soviet Union caused most Americans to quit worrying about the Bomb. This was actually a mistake; there are still many nuclear weapons in the world, and some went missing when the Soviet Union collapsed. We are now facing a global crisis as urgent as WWII, with some actors in it as fanatical as the Axis powers of WWII.

Answer 2

actually, it first impacted Japanese society ..
but seriously, it created a lot of fear, paranoia, and stupidity. It made us quite powerful and began the Cold War with USSR. It established solid justification for high defense spending and shifted the strategic mindset to deterence of nation-states. It also established great anxiety for global warming issues.