Even though there have been events that resulted in more deaths, I believe the timing of the two attacks and the novelty of the a-bomb must have created a deep scar in the collective memory of the Japanese. I have read that the Japanese monster movies, large impersonal monsters versus western monsters like werewolves and vampires, were a response to the bombings. Perhaps other things can be traced to Hiroshima and Nagasaki?
Although the proper answer to this question is highly opinionated, I will acknowledge that the bombings had a significant effect on Japanese culture; whenever an event becomes part of a national heritage, it is common to see it reappear in various forms of entertainment.
It is by this principle that I am not surprised to see the Japanese response to bombings that took place during a time of war over 70 years ago still echoing around. I personally am only distantly associated with Anime and Manga cartoons, but I realize that many of the plot points are symbolical of the nuclear attacks of 1945.
In a way, Little Boy and Fat Man did more for entertainment than Disney did; today's cartoons and film plots tend to revolve around the proper use of technology, human decency, uncontrollable power to destroy as well as the effects of incurable sicknesses. All these things are inherently related to the Atomic Age and are not solely Japanese either.
This article ties the development of Japanese cartoons to the historical significance of the bombings.
My lame-claim-to-fame for this response comes from personal contact with a Japanese national and a good friend of mine who has shared his thoughts about these historical events (although I do not presume they represent the whole of the Japanese public.) He has told me before that he feels no transcendent connection with the Japan of the forties. He said that after the War, Japanese culture evolved so quickly to recover a prominent position in the world, that the national mindset was scrapped and rebuilt. This is evident in the fact that Japan is today one of the most peaceful countries (Japan has denied and outlawed war-making powers through its Constitution) when historically the ideas of expansion and conquest were praiseworthy.
I would imagine the Japanese hold a unique position in today's world as being the only country with actual first-hand experience of the horrors that come with the use of atomic weaponry. If they sue for alternative methods in order to avoid a repeat of the bombings, I will consider their set of experiences when judging the balance between the good and harm that using a nuclear weapon may have in a conflict. One thing is for sure: that Hiroshima and Nagasaki will continue to influence the Japanese culture and, undoubtedly, the world culture.
Follow up question: how many film titles can we think of that have to do with atomic situations barely avoided?
Two key words in Japanese related to the Hiroshima and Nagasaki bombings are 被爆者 hibakusha, a survivor of the bombings, and its parent term 被爆 hibaku, meaning to sustain damage from an atomic bomb. This is homonymous with 被曝 hibaku, meaning radiation exposure in general, which gives the latter term a slightly gruesome overtone. At least to my ears. I do not know if Japanese hear it the same way.
In my opinion the most notable impact has been on the local culture of Hiroshima and Nagasaki. Hiroshima sees itself as a "peace city" and there has been a major anti-nuclear campaign based out of there, spearheaded by the successive mayors of the city. Nagasaki also uses the bombing as an educational opportunity to teach about the war in general.
It's not true that Godzilla represents Hiroshima and Nagasaki. Godzilla represents the Atomic Age in general and more specifically the 1954 Lucky Dragon incident.
Other cities in Japan were also destroyed, most notably Tokyo where over 200,000 civilians were killed in firebombing raids. Japanese political rhetoric and cultural attitudes towards war are derived much more from the war experience of all those who lived through that time, and less from Hiroshima and Nagasaki in particular.