It's hard to talk about counter-factuals ("what if" questions) of this sort, because they involve making huge assumptions about what might have happened had circumstances been very different.

But we can talk about what the advancement of US nuclear weapons technology did. It gave the US the ability to project huge numbers of nuclear weapons onto basically any place in the USSR. It allowed the US to use much more economical use of fissile material, allowing them to increase their stockpile dramatically. It allowed them to develop weapons that were small enough and light enough to put onto a wide variety of delivery vehicles, including ICBMs, SLBMs, and tactical weapons. It allowed the US to make weapons in the multi-megaton range, which reduced the accuracy needed by said delivery vehicles (the reason early ICBM warheads were so large is because they were so inaccurate), and it allowed them to threaten huge areas. It allowed them to create small weapons that could be dispersed in Europe and provide a threat other than that of total annihilation should military force be used.

So that's all something. None of the above would really be possible if they had frozen the work at WWII technology.

What didn't the weapons do? They didn't bring about absolute peace, they didn't cause all nations to immediately surrender, they didn't stop the Soviets from making their own nukes, they didn't stop the Soviets from expanding Communist influence into many counties, and ultimately they proved to be far less influential diplomatically than expected.

Did they deter Soviet aggression in Western Europe? Probably to some degree. Would the Soviets have invaded the continental United States if the US only had WWII nuclear technology? Probably not — invasion would have been a hard slog no matter what. But there are other possible negative outcomes other than Soviet invasion of the USA.