The tricky word here is "important." What does that really mean? Without a precise definition, this isn't an easy question to answer.

Sometimes when we use the word "important" in talking about history, we mean, historically causal. That is: would the world be significantly different if nuclear weapons didn't exist in the post-Cold War period? That's a large question, and not an easy one to answer. If you believe — as many do — that nuclear weapons "keep the peace" between mutually armed powers, then one can imagine all sorts of post-Cold War confrontations without them. If, on the other hand, you think their role in deterrence is exaggerated (that China has no interest in any kind of large-scale war with the US and would avoid such anyway, nukes or no nukes), then you might not. This does not exhaust the possibilities but I want to give you a sense of the difficulty of answering this question, much less answering it as a historian, given that it is in some way a question more about theories of political science than what happened.

This doesn't have to be our only definition of "importance." Perhaps one is asking: did nuclear weapons play a large role in national or international discussions? Even here we are plagued by vagaries. Large role... compared to what? And for whom? Certainly the question of whether North Korea was developing, or had, nuclear weapons, played a big role in US–Chinese–Japanese–North Korean relations. And the question of whether Iran is developing said arms has occupied quite a lot of diplomatic and even military planning with regards to the US–Iran–Israel–Russia. But it is hard to put something like this in perspective compared to all other potential issues in the world.

It is easy enough to say that they did not play a large role in US domestic politics, at least compared to their Cold War role. But even here one could ask what we really mean by that — all I mean by it is that voters have not made nuclear weapons one of their "issues," though the US has spent hundreds of billions of dollars on its nuclear arsenal in the years since the fall of the Soviet Union.

Anyway, this is not a comprehensive answer. I want you to see that this is a difficult way to phrase a historical question. A more targeted approach could be one of the following: "How did the US and Russian governments regard nuclear weapons in the post-Cold War?" "How did nuclear arsenals change in the post-Cold War?" "Was there a shift in thinking on nuclear weapons after the end of the Cold War, but before 9/11?" Things like that, which give one more to focus on and are less open-ended.