In addition to nshepperd's point, there's the fact that sending a bomber fleet to destroy an enemy city is very expensive - the costs of planes, fuel, and bombs add up quickly, not to mention pilots' lives. And if the defenders can destroy 50% of the planes before they drop their bombs, the bombing campaign becomes 50% less effective.

Whereas a strategic nuclear warhead only requires one plane to deliver it (or one ICBM). Much cheaper, much less risky, and much more stealthy. If you build a small fleet of nuclear bombers (or, again, a small stable of ICBMs), you can theoretically destroy all of the enemy's cities in one night.

If the atom bomb had by chance been developed a few years earlier, when the US still faced serious opposition in Europe, then quite probably they would have used it to wipe out all the major German and German-held cities.