Why would President Truman be against sharing the secret of the atomic bomb with the world?
Why would he support sharing atomic technology with Great Britain and only divulge minor details to the Soviets?
This issue is often described as "nuclear proliferation." Nuclear weapons carry the capability of mass destruction and very long term damage to the environment and collateral damage to any one downwind. Sufficient numbers of nuclear bombs are capable of causing a "nuclear winter" ending life on earth as we know it. This means that the fewer nuclear weapons there are in existence, the safer the world will be. Since the United States and the Soviet Union were not on good terms with each other, there would be no reason to give them information on how to build advanced weapons that might have been used against the United States.
You might also read the more recent news concerning worries about nuclear programs in Iran and North Korea -- issues of nuclear proliferation are still of major concern.