I agree with both of the previous posts: The Japanese would have eventually surrendered or otherwise succumbed to the Allied forces. The atomic bombings of Hiroshima and Nagasaki no doubt shortened the war and saved an incredible number of lives of soldiers from both sides who would later have been killed in the planned American invasion of the Japanese islands themselves. Without the surrender following the aftermath of the atomic bombs, Allied forces would almost certainly have continued their plan to invade Japan; casualty estimates were expected to be in the hundreds of thousands (some estimates predicted more than one million) on the Allied side alone. The Japanese soldiers' previous displays of their unwillingness to surrender would have signalled another long and bloody struggle within their own island borders. Many Allied veterans of the fight against Hitler in Europe were already awaiting orders for transfer to the Pacific; luckily for many Allied soldiers, the surrender following the dropping of the atomic bombs prevented them from risking their lives against a new enemy.

Agreed with the statements above, but let's also be clear.  The war would have ended the same way - with a Japanese surrender to the United States.  Consider the military position Japan was in by early August 1945, with no remaining air force or much of a navy to protect the home islands with.  Vulnerable to all other kinds of bombing as well, as they had been for a year or more. 

While historians consistently argue that the war ended sooner with a lower cost in lives, we will never know this for sure.  The war ended not because of the atomic bombs, but because the Emperor of Japan told the military to lay down their weapons.  Who is to say that the Emperor would not have done so after five more firebombings of Tokyo?  How can we accurately say what was in his mind in early August before the bombs were dropped, or how strong his will was by that moment?  We will never know, and at least part of why we argue that the bombs saved lives, in my opinion, is because we find a need to justify our actions as the only country ever to use nuclear weapons in war.

In my opinion, the ultimate outcome of the war (in terms of who won) would not have been any different.  Japan was defeated, even if it was not willing to surrender.

However, there would have been some major differences:

• Lots more people would have died -- American servicemen, Japanese servicemen and civilians.  The numbers would have been terrible.
• The Soviet Union would probably have taken all of Korea and invaded Japan.  This would have led to an all-communist Korea and most likely to a partition of Japan as happened with Korea and Germany.
• The occupation of Japan would have likely been more oppressive.  As it was, the occupation was very lenient to Japan.  But I think it would have been harsher if we had had to invade.  I wonder if Japan would have been able to emerge so quickly as a major economic power...