We declared War on America and Britain out of Our sincere desire to ensure Japan's self preservation and the stabilization of East Asia, it being far from Our thought either to infringe upon the sovereignty of other nations or to embark upon territorial aggrandizement.

But now the war has lasted for nearly four years. . .[and] the war situa-tion has developed not necessarily to Japan's advantage, while the general trends of the world have all turned against her interest. Moreover, 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 inno-cent lives. Should We continue to fight, it would not only result in an ultimate collapse and disappearance of the Japanese nation, but it would also lead to the total extinction of human civilization. Such being the case, how are We to save the millions of Our subjects; or to atone Ourselves before the hallowed spirits of Our Imperial Ancestors? This is the reason why We have ordered the acceptance of the provisions of the Joint Decla-ration of the Powers.

Aforementioned is a portion of The Imperial Rescript Ending the War (translated) which was drafted between August 10 and 14, 1945 by Sakomizu Hisatsune on the basis of Emperor Hirohito's words. The Emperor recorded it on the evening of August 14 and it was broadcast nationwide on the noon of August 15.

While the Emperor mentioned the bombings as the reason for surrendering when he addressed the civilians, in his surrender speech to the military forces, on August 17, 1945, he explicitly mentioned the Soviet Union as the reason. Following is a portion of the Surrender Rescript to Japanese Troops(translated):

 

To the officers and men of the Imperial Forces; Three years and eight months have elapsed since we declared war on the United States and Britain. During this time our beloved men of the army and navy, sacrificing their lives, have fought valiantly on disease-stricken and barren lands and on tempestuous waters in the blazing sun, and of this we are deeply grateful.

Now that the Soviet Union has entered the war against us, to continue the war under the present internal and external conditions would be only to increase needlessly the ravages of war finally to the point of endangering the very foundation of the Empire's existence.

With that in mind and although the fighting spirit of the Imperial Army and Navy is as high as ever, with a view to maintaining and protecting our noble national policy we are about to make peace with the United States, Britain, the Soviet Union and Chungking.

Apart from these two key events, the Hiroshima-Nagasaki bombings and the Soviet invasion, another argument which was first hoisted by Herbert Bix, who wrote biography of Hirohito focused on internal Japanese records, is that Japanese leaders were paranoid about an internal uprising in 1945, leftist and communist agitation being their main concern. 1940s brought about discontent because of rationing, bombings and other wartime exigencies and nationalism wasn't enough to sustain them. In the months before surrender, the Japanese government ramped up the amount of kamikaze attacks. Bix further argues that government saved face by claiming that they're surrendering in order to prevent further atomic bombings. This explanation justifies the contradiction between the emphasis on saving Japanese lives in Hirohito’s radio broadcast and the government’s previously supercilious behaviour towards the civilians.

With the shakiness of the evidence available, and Hirohito himself giving two very different reasonings to the different audiences, it is impossible to say for certain what caused the Japanese surrender and will probably remain a source of contention among historians.

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Sources:

•Hirohito and the Making of Modern Japan, by Herbert P. Bix

•Japan's Decision to Surrender, by Robert J. C. Butow

•taiwandocuments.org

See Did the dropping of the atomic bombs on Japan actually save lives? feat. /u/wotan_weevil and /u/restricteddata and /u/t-o-k-u-m-e-i

Copying and pasting from this thread a couple days ago.

This question comes up quite a bit here, probably because there is no consensus. /u/quiaudetvincet has a good answer here and searching the subreddit for "Japan surrender" also returns several other worthwhile answers. You can also find some similar questions in the FAQ and three older answers linked by /u/jschooltiger here.

Some well-respected historians believe that the atomic bomb was the primary factor, while others who are equally as qualified believe the Soviet Union's entry into the war was the primary factor. Others argue that the twin shock of both events happening in such a short period led to the decision.

Most everyone agrees that Japan had been seeking a negotiated surrender with the Soviets as intermediaries in summer of 1945, but only with specific terms. Even after the Soviets entered the war (with only a few hours notice) and the two atomic bombs were dropped, the Supreme War Council was divided on whether to continue fighting or not. It was a close decision, and thus it may never be possible to determine if one event had a larger impact than the other.