The Japanese were demanding terms before they'd surrender. The Potsdam Declaration said that the Allies would only accept "unconditional surrender".
The 4 terms that the Japanese were demanding were:
Term #4 meant that the Allies would be returning islands that were taken, such as Iwo Jima or the Philippines.
Term #3 mean that the Japanese would be returning to barracks and not surrendering by any definition of the word.
The Japanese knew exactly what happened in Hiroshima since they had their own nuclear weapon projects. They believed that since it took the Allies 4 years to make the first nuclear weapon, then it would take 4 more years to make the second one. We know this because we were reading their coded communications. The 3rd nuclear bomb (well, the "physics package", or pit, was) was in a plane flying from San Diego to Honolulu when the surrender was announced.
For a history that documented Japan's nuclear weapon program during WW2, I refer you to 2 books written by the same author: The Making of the Atomic Bomb and Dark Sun: The Making of the Hydrogen Bomb. After the collapse of the Soviet Union, a number of interesting documents about the nuclear programs of both Japan and the Soviet Union came out. Japan had 2 nuclear research centers, one in Tokyo and the other around Chosin Reservoir in what is now called North Korea. Battles around Chosin Reservoir were some of the fiercest of the Korean War.
If you want to learn more about the atomic bomb program in general, I cannot recommend these two books enough -- in many ways they cover the same stuff, but the first one is pretty short and will give you some background to make the second one, which is enormous, a little easier to stick with. My Dad worked in the 1960s for a bunch of physicists who worked on the atomic bomb (and I met a few of them as a kid), so I've always dug on atomic history.
Well, they may not know about radiation exposure effects (even though they know something about it. The absolute madman Louis Slotin took a dive near a reactor to fix it, and his colleagues were "shocked", and Japan itself tried to start a nuclear project, failing due to the fact, well, that only the United States had the capability of invest on it: developing nuclear bomb was expensive as hell), but they surely knew that it was different. They saw just a couple of planes, and then hell broke loose, and the aftermath of the explosion was a bit worst than the one of a firestorm. Mind that a fire bombing is not meant to create a shock wave as an atomic bomb does.
And after all of this, after the bombing of Nagasaki, the emperor (I won't ever remember his name) stated:
"The enemy has begun to employ a new and most cruel bomb, the power of which to do damage is, indeed, incalculable, taking the toll of many innocent lives. Should we continue to fight, not only would it result in an ultimate collapse and obliteration of the Japanese nation, but also it would lead to the total extinction of human civilization."
"a new and most cruel bomb": average Japanese may not care about the difference between a firebomb and an atomic bomb, but I can assure you that upstairs, they were concerned about the use of the new weapon.
As an ending note, if you love to read, and if you don't care about lengthy readings, Richard Rhodes wrote a couple of very well documented books about the matter:
- The Making of the Atomic Bomb, which is about the development of the weapons, the Manhattan Project, and ends with the Nagasaki bombing;
- Dark Sun: the Making of the Hydrogen Bomb, which is an amazing book that sometimes resembles a spy story and sometimes is just plain scary. Castle Bravo Fuck Up, for example.
Have a good day :)