In the chapter of Traditions & Encounters that you appear to be referring to (Chapter 12 in my brief Second Edition), Bentley and Ziegler are not really discussing differences between Korean and Vietnamese society per se. Instead, they are discussing the different ways in which Korea and Vietnam responded to Chinese influence.
China was by far the most dominant country in East Asia. It was powerful and rich. Because of this, it was able to spread its influence across the region, including what are now Vietnam and the Korean peninsula. While Korea and Vietnam both came to be influenced by China, there was one major difference that Bentley and Ziegler identify in their respective responses to China. The major difference was that Korea essentially accepted Chinese domination while Vietnam did not.
In Korea, China came relatively easy to a political compromise. The Chinese left the Silla kings in place, but those kings acknowledged Chinese supremacy. They brought gifts to the Chinese emperor and regularly renewed their vow of loyalty to him. This was very beneficial to Korea in economic and social terms. By contrast, the Vietnamese were much more reluctant. They did give tribute to China at times, but they also rebelled. This culminated with them throwing off Chinese rule and becoming independent in the tenth century.
Thus, the major difference between these two countries (in this chapter of Bentley and Ziegler) is that Korea was much more willing to accept Chinese domination.