Yes, the US may possess more advanced weaponry, but manpower is a different story.
If they fought against both the PRC and the USSR, their sheer numbers might be able to overwhelm the US. China had about 600 million people at that time, while the USSR had about another 250 million or so, while the US only had about 150 million people. That's about a six and a half to one advantage, and the USSR and PRC also enjoy the advantage of a shared border.
Also, Stalin once brutally calculated that the Nazi invasion of 1941 was the equivalent of fifteen atomic bombs, so the USSR could survive a pre-emptive nuclear strike by the US. Considering the sheer landmass of the USSR, along with China, there simply wouldn't be enough targets to bomb.
Mao also calculated that even if 300 million Chinese were to die, there would still be three hundred million Chinese left to do the job. Also, if the Japanese couldn't conquer China while it was at war with itself, damned if I see how the US could once China was united under communism. I think you would need at least 15-20 million troops to keep China pacified, and I don't think the US had that kind of numbers.
So all in all, I think WWIII would have lasted even longer than WWII, possibly not ending until sometime in the 1960's, and I don't have any real idea on how it would end.