My grandfather asked me this question recently and it occurred to me that I don't know the answer. If it's a vague question to answer, the analogy I would use would be that France and Germany became culturally distinct (or at least started to be) after the fall of the Carolingian empire which separated East and West Francia.
Do China and Korea have a similar event in their history which marks their divergence into separate cultures rather than Korea just being a province of China?
u/LondonSeoul is right. As with Japan and Vietnam, Korea originated as a separate civilization from China. The earliest Korean kingdoms were based in the Korean peninsula, while their Chinese contemporary states originated around the lower Wei River and middle Yellow River basins, several hundreds of miles west of the Korean peninsula. The first Korean kingdom of note, Gojoseon dates to the first millennium BCE, during the pre-Imperial Zhou Dynasty era in China. At this point, the Zhou had spread across the North China Plain from the Yangtze River Delta in the southeast to the Bohai Sea Rim in the northeast. China didn't expand to and annex parts of the Korean Peninsula until 108 BCE, during the Han Dynasty.
Source: Cartwright, Mark. "Ancient Korea." Ancient History Encyclopedia. Ancient History Encyclopedia, 21 Oct 2016. Web. 01 Jan 2018.
If you really need to point to an event, the conquest of the last of the Han Commanderies by Goguryeo in 313 CE would be the date after which China never again had direct control of the Korean peninsula for any significant length of time.
The question is based off a false (albeit very common) pretense though; psyche_da_mike kind of establishes this, but to emphasize, Korean civilization, while immensely influenced by Chinese throughout the millennia, was from its incipient stages in a distinct cultural sphere from the Central Plains cultures that would make up the core of Chinese civilization; they were never non-distinct civilizations.
The period the Chinese cultural influence was most dominant was following the Han Dynasty's conquest of the ancient peninsular kingdom of Gojoseon in 108 BCE (which psyche_da_mike mentions), after which it held direct control of a portion of the peninsula (with influence over the rest of it, as well as some of Japan) for around 400 years; there's been a lot of debate over how directly/influential Chinese rule was during this time (and it varied over the period too), but the second to last study in The Han Commanderies in Early Korean History makes a compelling case that those regions of the commanderies were fairly thoroughly integrated into the Han empire, and had an enormous cultural pull over the peninsular peoples. Even then though (again from studies in that book), outside of the highest levels of governance, the administration was still generally carried out via local indigenous chiefs rather than Han Chinese officials, and despite a lot of immigration by Han Chinese people (both before and during the commandery period), the local culture would be an indigenous-Chinese mixture rather than being totally Chinese (and it's worth mentioning, even in China proper at the time, there was enormous cultural variation).