Which rivalry, specifically, are you referring to? Just in the closing days of the cold war, or do you mean through to today? If the latter most of that will be out of the scope of the 20 year rule on AskHistorians.

US-Soviet relations did thaw under Gorbachev, and the mid-to-late 1980's saw a dramatic reversal in tension from the dangerous levels maintained in the early 1980's. The decision made by Gorbachev to not militarily intervene in the affairs of the Soviet satellites in Europe allowed for the revolutions of 1989 to take place. The loss of the "Soviet Empire" and the eventual dissolution of the USSR signaled the final ending of the Cold War. US-Soviet relations improved, the Soviet Union no-longer had notable influence in Eastern Europe, and internal economic problems came to the forefront of the Soviet, then later Russian agendas. The 1990's were the highest point of US-Russian relations in modern times, because the Russian state was both unwilling and unable to flex it's muscle internationally, while the United States was willing and able to help Russia with some of it's domestic problems.

The Russian economy began improving near the end of Yeltsin's term in office, and the recovery peaked under Putin. We are now beyond the scope of the 20 year rule, but the point I'm trying to make is that a necessary precondition of a resurgence in rivalry was this economic recovery. Although the lion's share of tensions did occur after Yeltsin left office, there were still hints of things to come in the early-mid 1990's.

One of the big open questions after the end of the cold war was the status of NATO. Although few people would push to outright abolish the alliance, the former Warsaw Pact nations of Eastern Europe were pushing heavily for membership, and the existing NATO members all eventually came around to admitting them. This frustrated Yeltsin, who made his objections known to Clinton. Despite this, Yeltsin and Clinton did hold successful summits and agreed on a wide variety of issues, notably Arms Control. There's a contemporary news source for the 1997 summit available here. Yeltsin, ultimately, was more interested in making progress in other areas and cementing good US-Russian relations than he was on bitterly fighting against something that he knew he could not stop. As noted in the article, Yeltsin's political opponents minced no words when criticizing him over this attitude, and Yeltsin's eventual successor, Vladimir Putin, would eventually take a much more assertive position in Europe.