It's important to note that while it was always framed as an ideological struggle (Communism vs. Capitalism), in reality the lines were drawn more around power than strict beliefs. This is not to say that beliefs didn't matter (e.g., it was largely differences in beliefs that led to the Sino-Soviet split), but they can be overestimated. If the USSR was a small country that didn't, in the eyes of the US, threaten the international (US-led) order, would the Cold War dynamic have existed in the same way it did? Probably not. Did Cuba or Vietnam, for example, threaten the US except as a perceived extension of Soviet power and the possibility of that power expanding further? Not really. Both the US and USSR showed the ability to make alliances with countries that didn't quite meet their ideological litmus tests, if they thought it was in the greater interest of counter-balancing the power of the other superpower.

Cuba, Vietnam, Laos, China and North Korea all originally interested the US because they were seen as extensions of Soviet power. The interesting case here is China, which as of the 1960s split from the Soviet sphere of influence and went its own way. By the 1970s, Nixon had "re-opened" relations with China (as a realpolitik wedge-driving strategy), and by the 1980s the Chinese had so modified their economic model as to be only debatably "Communist" (they are some kind of hybrid model).

With the fall of the USSR, the situation changed. Now Cuba, Vietnam, Laos, and North Korea were effectively without a superpower patron. In the case of the first three countries, that did lead (eventually) to something like a normalization of relations (the slowness of Cuba was less about its ideological standing than about the powerful influence of Cuban-Americans in the US political system, especially in Florida). North Korea, by contrast, became even more isolated, and was pulled into a complex multi-party relationship with China, South Korea, Japan, and the US.

Of these, China is the only real "successor" to the USSR in US discourse or economic activity, but its relationship with the US is very different — not quite a Cold War, but also not quite peaceful. The two countries are economically highly entangled, even though they are often at odds with each other on key political issues. If you wanted to argue that the Cold War didn't end, then saying that it moved from the Soviet sphere to the Chinese sphere is a plausible argument, and indeed has been made (you could also call it a "New Cold War").

Of course, if one is really not concerned with ideology, then one might ask: did Russia just end up taking up the legacy of the Soviet Union? Maybe, maybe not — the Russian Federation is not nearly as powerful as the USSR used to be, but is still a major global force and has the power (as we have seen) to do a lot of foreign damage. So again, one can argue if the Cold War "ended" and whether it just changed — instead of the appearance of bi-polarity (US vs. USSR) you have something like multipolarity (US vs. Russia vs. China), or something else entirely (some have claimed unipolarity, e.g., US vs. the world).

It is worth noting that scholars of the Cold War today are generally somewhat downward-looking on the idea that the Cold War itself was really as bi-polar as it is (and was) often talked about. Yes, the US-Soviet conflict set many of the tones and tensions of the period, but there were still alternative sites of power and contestation outside of them, which often gave other countries room to maneuver. It is impossible to talk about, say, the actions of France, West Germany, India, Egypt, Yugoslavia, Taiwan, and Japan without talking about Cold War tensions, but at the same time these countries were at times very astute in manipulating the US and Soviet Union in achieving their own aims (or declaring themselves entirely separate from them, such as with the Non-Aligned Movement). Much modern Cold War scholarship is about the complexities of these superficially "minor" negotiations and the way in which they actually caused "major" movements among the superpowers.

Lastly, it is worth emphasizing that "the Cold War" is not a natural category. Even something like "World War II" had have is origin and ending points somewhat disputed, but the Cold War, as a tense relationship rather than an overt conflict, is even more slippery. When did it start? (1945? 1947? 1949? 1917? 1922? Earlier? Later?) When did it end? (1989? 1991? Was detente a "break"? Were the early 1990s a "break"?) We can debate these questions, but it's important to recognize there isn't some self-evident answer: if we offer up a new argument for when it started or ended (if it did), we need to always ask, "what are the stakes of this periodization? why have it at all? what does it shed light on, and what does it miss?" Because we're necessarily describing an interpretation of history when we deal with any kind of categorization like this, and there are ways all interpretations (and periodizations) can do benefits or damages to our understanding of the past.

The Cold War was a conflict between two superpowers. Once one of them fell, it ended. A new cold war could arise if a new superpower rises to defy the US.

Now this is an explanation of why the Soviet Union fell:

Lots of reasons, the main ones being economic and nationalism.

From the economic viewpoint, communism doesn't grow as much as capitalism. This is seen everywhere. The economic miracles of Vietnam and China more recently were done by adopting capitalist reforms (especially freedom of price setting), just like Soviet Russia under the NEP (New Economic Policy) in the last years of Lenin rule. The decelaration in the rate of growth seen in the Soviet Union in the 70s and 80s, called "Brezhnev stagnation" by economic historians, is an important factor in its downfall. There are many explanations for it, but most economists that I've read (like Acemoglu and Robinson, for example), stress that the dictatorial planned economy that the Soviet Union used could work for a short while in attaining high rates of growth, but not on the long run. On the short term it could deliver high rates of growth in the first 5 year plans because the technology was not that groundbreaking (it was mostly just basic infrastructure, like power plants, steel mills, education, etc, that Russia lacked). But when the Soviet Union achieved a certain level of development, it couldn't grow its productivity more by investing in basic infrastructure, it had to invest in more complex things, and this is where planned economy failed. The lack of incentives that free market, free pricing, bankruptcy and free accumulation of capital have caused the Soviet industries to become more and more inneficient, which was only made more problematic by the systematic corruption in the Russian society (which still exists today, it's not just Russia under communism that had it).

As the rate of economic growth became slower, the Soviet Union faced more and more problems maintaining its arms race with the West (and the arms race got way more expensive during the Reagan admnistration in the US, as he was much more aggresive towards communism around the world). It cost more and more of the government budget to do it, and there was less to be given to the people as consumer goods. This was made more problematic with the Afhganistan War (wars are expensive) and the fall in the price of oil that happened during the 80s. The Brezhnev stagnation could be masked during the 70s because the Soviet Union discovered and exploited lots of oil during the oil shocks of the 70s, becoming one of the most important net exporters. But as the price of oil fell as the shocks subdued, the Soviets couldn't use that revenues anymore to fund their wars, arms race or investments. The West, which was mostly a net importer of oil, on the other hand benefitted from oil price glut. This increased the difference in the rate of growth and when the press censorship was diminished in glasnost people started to realize how worse they lived compared to the West.

Glasnost also caused problems with nationalism. Nationalist pressures that were kept contained due to political repression started surfacing again. At first these happened in the Caucasus, the Baltic republics (which had been invaded in the 1940s, with many people still remembering how it was to be independent) and Central Asia. Gorbachov, the reformist Soviet Union leader, didn't repress the nationalist leaders, as his predecessors did. He also famously did not do that when the Eastern block countries tried to do things differently. Previously when that happened (Hungary in the 1950s, Czechslovakia in the 1960s) Soviet/Warsaw Pact troops came to repress it, but Gorbachov didn't do it (this was called the "Sinatra doctrine" due to a famous song about letting people go their own way). The minorities in the Soviet Union started to see the possibility in getting more rights or even freedom. As this happened, so did the majority (Russians) started to realize that they didn't have some things that the minorities had, like a parliament of their own. This is very important, because it is when Boris Yeltsin started to appear as a leader of the Russian people, just as there appeared leaders of the other ethnicities in the Soviet Union. If one could compare, it would be like English people in the United Kingdom demanding their own parliament, because Scots and Irish have their own.

In 1992, as the Eastern Block had basically disappeared, Gorbachov made some moves to appease the Soviet hardliners. He appointed a hardliner as his vice president and in the Congress, and then went on a vacation. The hardliners then staged a coup to try to undo all his reforms, keeping Gorbachov imprisioned in his dacha (summer house). But the Soviet people opposed that, one of their leaders being Yeltsin. Gorbachov was put in the sidelines (literally he didn't move from his dacha), as the nationalist leaders started negotiating between themselves what would happen next. In December 1992 Gorbachov finally resigned as leader of the Soviet Union because the nationalist leaders were already creating their own treaties (CIS) and he had no power left at all.