There is a bit of a difference between needing to commit dozens or hundreds of expensive bombers to kill a city, and only needing to commit one. You go from being able to knock out a city in a day (... and with, probably, weeks worth of munitions production) with an air wing, to being able to knock out all the cities in a country in a day with an air wing. With enough nukes and a modest number of bombers at one's disposal, all but overwhelmingly-strong air defenses become incapable of protecting a country from destruction—enough will get through to cause catastrophic damage (talking 40s and 50s tech here—as air defense tech got better, ballistic and cruise missiles overtook bombers in this kind of scenario).
Concentrating the damage like that gives the enemy far less time to adapt or rebuild, and less capacity to do so—you can out-produce heavy damage that's spread out over a long period of time, you can adapt to it, you can shift tactics and resources around to reduce the rate of damage in the future, but such efforts are hopeless against nukes.
It wasn't especially in-play during WWII since even the US couldn't (yet) build the bombs fast enough for them to really matter that much versus what could be done with conventional weapons, but it did matter soon after as stockpiles began to grow and production rate increased. Nukes represented a huge multiplier in the amount of destructive power a state could realistically stockpile and deploy in short order.
Concentrating the damage like that gives the enemy far less time to adapt or rebuild, and less capacity to do so—you can out-produce heavy damage that's spread out over a long period of time, you can adapt to it, you can shift tactics and resources around to reduce the rate of damage in the future, but such efforts are hopeless against nukes.
It wasn't especially in-play during WWII since even the US couldn't (yet) build the bombs fast enough for them to really matter that much versus what could be done with conventional weapons, but it did matter soon after as stockpiles began to grow and production rate increased. Nukes represented a huge multiplier in the amount of destructive power a state could realistically stockpile and deploy in short order.