The subsequent decision of the Czechs and Slovaks to go their separate ways, which was implemented in 1993, was equally peaceful. Therefore, it was inevitable that it would be labeled the "Velvet Divorce".
As I mentioned here, the reunification of Germany was directly tied to the events of 1989. The division of Germany that existed between 1949 and 1990 was completely based on Cold War tensions. The USSR could not come to agreement with the other occupying powers, the U.S., Britain and France, on arrangements to unify the country, which had been the goal when the occupation zones were mapped out at Yalta in 1945. But, once the Cold War was over, the dividing line was quickly erased.
By contrast, the multinational state of Czechoslovakia came into existence long before the Cold War started. As was the case with Yugoslavia, Czechoslovakia emerged out of the collapse of the Austro-Hungarian empire in World War I. There is no logical reason why the Czechoslovak state could not have been kept together post-1989, if the political will had existed for that. But it did not.
The country's Communist dictators had no incentive to split up Czechoslovakia, because that would have diminished their power. However, after 1989, democracy allowed interests in both sides of the country to push their separatist agenda. In other words, the grass roots had a very different view than the former top-down rulers. The democratization allowed Tip O'Neill's famous saying to apply: "All politics is local."
To summarize: the Velvet Divorce was tied into the events of 1989, just not as directly as German reunification was.
But why wasn't the breakup of Czechoslovakia as violent as that of Yugoslavia? Here is a summary of a lecture by Valerie Bunce, a Cornell professor, going into at least as much detail as you're probably interested in, regarding differences in governmental structure. The key is:
Serbs were angry at not getting what they saw as their due, and Serbian leaders had significant institutional resources at their disposal to give a clear voice to these resentments.
Serbia's Slobodan Milosevic, a product of Yugoslav Communism, had, by the early 1990s, no counterpart in Czechoslovakia. As was typical of Communist leaders, Milosevic gave top priority to preserving his power. By contrast, the democratic leaders of Czechoslovakia had no incentive to use force to prevent a breakup.
Aside from those issues of governing structure, the religious demographics are different. Roman Catholicism is the primary religion in both the Czech Republic and Slovakia. That differs markedly from the combination of Catholics, Serbian Orthodox and Muslims in Yugoslavia. That seems to have been a major factor in the animosity between the various Yugoslav nationalities.