How did the early development of Vietnam compare and contrast with that of Korea? What were the similarities and differences in their relationship with Chinese governmental and cultural influences?
Your answer all depends upon how far back "early development" goes to your mind. Archaeologists say the Indochina Peninsula has been inhabited since the Paleolithic era, and the Vietnamese culture goes back to the Neolithic era. Multiple migrations populated the area. The Vietnamese language reveals the identity of the migrants as it combines elements of the Mon-Khmer, Tai, and Austronesian languages. Vietnamese also contains borrowed Chinese words that relate to philosophy, government, and literature; these are from later Chinese periods.
The spread of Buddhism from China into Korea and Viet Nam was a major cultural influence emanating from China and affecting those two other cultures. A very clear and concise discussion of the question you raised can be found here:
The final paragraph provides a succinct summary:
Chinese culture spread to the sedentary agricultural populations of Korea, Japan, and Vietnam in the first millennium C.E. Chinese writing, bureaucratic organization, religion, and art all made impressions on the indigenous cultures. In general, the local elites of the three regions actively sought to emulate Chinese models. Differences within the three areas resulted in divergent outcomes and alternative mixes of the indigenous and the imported. China was able to establish direct control over Korea. In Vietnam, Chinese influences mingled with Indian cultural contributions. Only Japan remained permanently independent of China and, thus, was able to selectively adapt Chinese models to Japanese needs. In East Asia, as a whole, cultural exchanges took place in isolation from the rest of the civilized world.
Korea, and Vietnam too to some degree, have very clear cultural identities that they have built social constructs around to protect it. Koreans are a very exclusive society and so are Vietnamese. You can tell by the way they have both treated mixed race children in their countries, not out of cruelty, but because that preservation construct is so very strong.
We must recognise, as #2 points out, that Korea is much more fiercely independent, and, whilst it does owe a lot to Chinese culture, defines itself as being profoundly separate from China and has a very strong sense of self-identity. Korea has more of a history of being oppressed by China which may have helped it to develop this fierce sense of independence.
These were pretty similar. Both Vietnam and Korea got most of their "higher civilization" from China. That is, they got things like their writing system, their system of government, and Confucian ideas from China.
Korea was able to stay more independent of China, though. This was because it was geographically more separe and because its native language was not as similar to Chinese. These things made it so that Korea kept more of its own identity than Vietnam did even though both borrowed heavily from China and accepted Chinese political control over them to a great degree.