The atomic bomb is often talked about as an entirely separate weapon of awesome destruction, rather than as the ultimate extension of the strategic and terror bombing used throughout World War II. Discussions about the use of the bomb and whether it was justified often center on how powerful it was and the immense destruction it caused, not always with an understanding of the strategic bombing practiced throughout the war.
But was the devastation inflicted on Hiroshima and Nagasaki really different than the destruction of cities by conventional firebombs, like Dresden or Tokyo? Did leaders at the time view the bomb as an extension of strategic bombing, or as something in a class of its own?
The American planners believed the atomic bombs to be distinct enough to require their own planning, procedures, and authorization. They had conversations about the atomic bombs that they did not have about strategic bombing. That is partially an artifact of the fact that the people who were in charge of the atomic bombings were not the same people in charge of the strategic bombing campaigns — you didn't have input from as many civilians (scientists) in the latter, you didn't have that coordinated from the political wing of the American government, but from the military itself. And this is partially a difference caused by the different potential seen in the atomic bombings: it was not just an attack, but an inauguration of a new age, in the minds of the people using it, and they were as concerned about the postwar implications as they were the actual attacks themselves. So they were deliberated upon at some length. There was almost no civilian authority behind the strategic bombing campaigns, whereas civilians (notably Stimson and Truman) were involved in the selection of targets for the atomic bombs (notably the question of whether Kyoto would be a target).
In terms of the target planning, the atomic bombing targeting was more deliberative in many ways. They had somewhat elaborate criteria for what they wanted out of a target, because they were only planning to use a few of these bombs at most, and wanted each to be a spectacle. This is not the case with the firebombing, which was much more structured around the intended damage to industry and so on.
In terms of the actual devastation, there were differences, though from an ethical perspective they are pretty similar. The atomic bombs were deadlier to their targets because of their lack of warning and the speed in which their effects were output. Firebombing was relatively slow (it took hours, sometimes days to destroy a city), and could be detected in advance (you could tell when hundreds of B-29s were approaching; the sound alone was a give-away). Even at Tokyo the number of people killed per area of city damaged was a lot lower than at Hiroshima or Nagasaki, and Tokyo was by far the worst because it came as a surprise (after Tokyo, the Japanese got better at evacuating cities) and because it was so population-dense (it was one of the highest-population cities in the world, so the numbers are very large even though the percentage killed is a lot lower than Hiroshima). I have written on this at some length here. This is separate from the long-term effects like radiation exposure, which were not taken into account when the atomic bombs were planned (and may be given too much emphasis; they make up a much smaller number of the casualties than most people realize).
So I would say: there is evidence to suggest that the atomic bombs were considered different by those at the top of the political hierarchy than the other strategic bombing campaigns. That is not to say that they were entirely discontinuous, or that everyone shared this view (the military men, and those lower on the totem pole, tended to see them as continuous). And their effects are somewhat similar, and but with key differences. Emphasizing either those similarities or differences has long been part of the moral arbitration of the atomic bombings, though I don't think it gets one very far — both enterprises involved massacring civilians by the tens of thousands, and I don't think that makes either of them morally excusable, personally.
The real place where one sees the jump in rhetoric is not in the specific deployment in World War II, but what people thought the implications were — what the long-term issues were. Already in the spring of 1945, they knew that the atomic bombs they were preparing for the war were but preliminary versions of what might be perfected with a little more time and work, and were talking about multi-megaton possibilities (even 100 megaton weapons!). They were thinking not only in terms of a handful of bombs but arsenals full of them; they were imagining the possibility of species annihilation. These imaginations are what made them treat the atomic bomb differently, not a real perception that the effects were so different from conventional weapons. For more on these, see Spencer Weart, Nuclear Fear.