Comparing modern Taiwan and modern China is an interesting and important exercise. However, when I answer this question, I am going to talk about more than just the Cultural Revolution because many changes on the mainland were the result of several decades of government oppression, not just the Cultural Revolution years. Despite liberalizing the economy, Deng Xiaoping posthumously justified Mao's trust in him by continuing the cultural and societal oppression; it was more open under Deng than it has been since, but the country was never going to turn into a democracy under Deng, and the Tiananmen Square incident in 1989 is really pretty solid proof of his loyalty to the Party.

So, to start out, yes, the Cultural Revolution changed (I hesitate to say "destroyed", only because even if modern culture on the mainland differs a lot from the Chinese culture of the late imperial or even Republican periods, the people are still Chinese and they still have a culture) Chinese culture on the mainland quite drastically. But again, it was more than the CR that was responsible. In terms of material culture, frankly, a lot of things were destroyed during the CR. Temples were attacked in particular as the many young people who were the main participants rushed to replace the "olds" with the "news". However, this goes deeper than just destruction of temples. Melissa Brown, an anthropologist specializing in Taiwan, but who has also done significant work on China (particularly in order to compare the two places), has written a not-insignificant amount on the destruction of folk cultural practices during the CR. Brown uses Hubei, and particularly Enshi, to discuss how the identity of the Tujia people was crafted by the CCP as "non-Han"; in fact, the ethnic group was "created" by the CCP out of a people that had always considered themselves Han. In any case, the CR led to local geneaologies being lost. The folk religion suffered greatly under the CCP as the Party went to great lengths to restructure and re-educate the Chinese populace. This was a nationwide movement. Beginning in 1957, with the Anti-Rightist Campaign, and continuing through the Great Leap Forward and Cultural Revolution, the government began cracking down on traditionally Han practices. The focus has been on religion in particular, including Confucianism, which is no longer really an important belief system on the mainland, particularly by the government, which mostly uses the term as a political tool. As Robert Weller has pointed out in his research, the practice of the folk religions (by far the biggest religious tradition in both China and Taiwan) was more or less eradicated, and this extened to things such as mortuary practices. The attack was thorough. It's only been in recent years that religion has made a little comeback, but even so it does not have the same place in society as it does in Taiwan. In addition, minority Chinese languages (such as Cantonese, Hokkien, Hakka, etc.) have been the focus of governmental crackdown in favor of Mandarin being "the" Chinese language. China in many ways is searching for an identity in the wake of many of its traditions being eradicated.

Taiwan never suffered these kinds of crackdowns. The Guomindang was certainly an oppressive regime and severely limited people's rights, but it did not interfere very much with these traditional Han practices. There was still an emphasis on patrilineal descent, and in fact the folk religions were not really messed with at all. Basically, if you allow me to summarize a bit here, Taiwan went through gradual cultural changes over time and traditional practices were never eradicated. Religion still plays a major role in Taiwanese culture today. In China, there was a lot of turmoil and trauma regarding cultural traditions being targeted for many decades by the government.

Something else I want to talk about since it is a common bit of racism basically everywhere that I hear or read. That is the belief that Chinese culture is somehow predisposed to authoritarianism. This, frankly, is a load of BS and racist BS at that. Most of the world for most of history was either authoritarian or at least pretty far from being democratic. You can argue that literally almost any culture is predisposed to authoritarianism; this is not a Chinese-specific phenomenon. And, in fact, it has been argued that many aspects of traditional Chinese culture helped lead to democratization in Taiwan, and the continued liberalization there. Cheng-tian Kuo has also found that adherents to the Daoist and folk religions tend to favor pan-green (so, the Democratic Progressive Party, the more socially liberal party) candidates over pan-blue (the Guomindang, the more conservative party) candidates. Together, these groups are the largest in Taiwan. Additionally, people who identify as "non-religious" actually do usually worship deities and are really pretty much adherents to the folk religions. Traditional religion in Taiwan has a had varying effects on politics there, whereas on the mainland it has been government trying to eliminate religion altogether. In any case, it seems ridiculous that a country that is generally recognized as being "more" traditionally Chinese than the mainland is forgotten in the haste to describe Chinese culture as oppressive. And this isn't even bringing into the discussion the existence of Japan and South Korea, two more very democratic states that also arguably have more living traditional Chinese culture than the mainland.

Sources:

Is Taiwan Chinese? The Impact of Culture, Power, and Migration On Changing Identities by Melissa Brown

Alternate Cvilities: Democracy and Culture in China and Taiwan by Robert Weller

Religion and Democracy in Taiwan by Cheng-tian Kuo

Mao's Last Revolution by Roderick MacFarquhar and Michael Schoenhals