This is a complex topic, but I can generalize and simplify it best as, Korea and Japan took cultural cues from Ancient China but their culture eventually diverged as their countries became more developed and unique.
China was the most influential state in East Asia for most of its history. The biggest change to Chinese culture occurred after the fall of the Qing Dynasty – the establishment of the Republic of China in 1912 and after that the People’s Republic of China in 1949 eliminated or lessened a great deal of its ancient customs including the influence of Confucianism and Buddhism in everyday society. It also developed its own character set, Simplified Chinese. Modern day China today operates as a single party controlled capitalist state – the closest country that can be compared is modern day Russia – which is for all intents and purposes also a single party controlled capitalist state. It retains a semblance of its ancient culture but only minimally.
Ancient Japan (Yamato) took a great deal of its customs (including Confucianism and Buddhism) from Ancient China but especially Tang China / Song China – after the Mongol Yuan dynasty took hold and relationships with Japan deteriorated, Japan mostly isolated itself after the Tokugawa Shogunate took hold. Japan still uses traditional Chinese characters (Kanji) in addition to its own character sets today. Modern day Japan is more or less its own culture which remains quite distinct from modern day China.
Ancient Korea (Silla/Goryeo/Joseon) took a great deal of its customs (including Confucianism and Buddhism) from Ancient China but especially Ming China with which it had a great relationship – but after the Manchu Qing Dynasty took hold – relations deteriorated and Joseon mostly isolated itself as the ‘Hermit Kingdom’ and developed its own culture. Korea used to use traditional Chinese characters (Hanja) but its has almost all been replaced by Hangeul (the Korean alphabet) today. Korea was colonized by Japan in the early 1900s and a lot of Japanese culture – including the strict hierarchical society and ‘chaebol’ work culture – was adopted by Korea.Today’s Korea is a mix of Ancient Chinese culture and modern Japanese culture. North Korea remains more or less, a smaller version of Mao-era China.
As a bonus, I’ll thrown in Taiwan as well.
Taiwan was minimally populated in the Qing dynasty but became more developed after Japan took control of Taiwan in the late 1800s – after the PRC won the Civil War the KMT fled to Taiwan and took with it the remnants of Ancient Chinese culture/ROC culture with it, including the use of Traditional Chinese characters. Today’s Taiwan is a mix of ROC culture and Japanese culture.
Obviously this is all very high level but I think it provides a nice general overview.