In the short run, things would go better for Europe. There would be no three-way standoff with a wounded Stalin pouring resources into armaments and Germany turning into a hermit state. But in the long run this would have horrifying results. We know today that only the balance of terror has kept Germany from running wild. A world with one nuclear-armed nation can not exist in stability. America would find it impossible not to dominate the globe militarily (rather than economically as it does today) and the outcome would be a world war uniting all other powers against Washington.
OTOH, if the US got the bomb first, would it have chosen to prevent the USSR from developing their own? It would, after all, have required an insanely expensive military occupation to prevent it. A USSR which was part of a winning alliance rather than having an insanely genocidal enemy within 5 minutes missile-flight from their capital probably would have been a much more civilized place than the ultra-paranoid, ultra-militarized state of OTL.
We probably could see post-war a fairly peaceful coexistence: surely the Soviets would have been glad to concentrate on reforming their economy and raising their living standards rather than engaging in a pointless arms race. Sure, they probably would have required a sphere of influence of sorts in eastern Europe (judging from wartime talks before the first German nuclear strike), but this would have been better for the Romanians and the Bulgars than their role of brutally exploited German puppets, and of course, it would have been much better for the Poles.
A more open and less financially strapped USSR probably would have reformed it's economy a lot earlier than OTL, too: the near-collapse of the 90's would have been avoided, and we wouldn't be propping up the USSR economically nowadays to help contain the Germans...
Thought: in this alternate, does Stalin still grab Korea and N. China if he doesn't need "compensation" for his losses? Do we see a united Guomindang China or do the Chinese communists still manage to carve out a territory of their own without the help of the Red Army?
CalBear, I think you exaggerate the threat of biowarfare: it's not like anyone but the Big Three (and perhaps the UK) have the tech and the inclinations to do any bioengineering, or develop reliable delivery systems. Sure, lots of dictators now have their vats of Anthrax and God knows what else, but they're really more a threat to their own citizens than to anyone else. I mean, during the last Indian-Pakistani brushup, the same bioweapons release (and I say we _still_ don't know who did it) killed almost as many Pakistanis as Indians (or the other way, killed more Indians than Pakistanis) - not to mention the Afghans. And most of these threats can be handled by good public health systems (crazy thought: without the threat of biowarfare, would the US have no universal healthcare system? Naah, too crazy.)
Bruce