Your terminology is a bit off here. "Daewang" means "Great King"; "Hwangje" means "Emperor."
China was regarded in a similar manner to the way Europeans regarded Rome and the Pope. Not to say that the Emperor of China had the same status as the Pope, but he was representative of a civilization that people in that part of the world considered the bastion of civilization.
That the Koreans referred to their rulers as "King" also has to do with tradition; Korean tradition regards the first mythological rulers as the "Supreme King." Until the Korean Empire period, Koreans, unlike the Chinese, believed that the title of "Emperor" was reserved for Heaven itself. The Goryeo period rulers did use Imperial terminology, but still did not refer to their ruler as "Emperor" because of this tradition. The mythological founders of Korea were not considered "Emperors" becasue they were considered "representatives" of Heaven. On a side note, even after independence, Vietnamese kingdoms regarded China's supremacy, but locally refferred to their own rulers as "emperor"; basically, naming is merely political lip service.
The use of "Hwangje" (Emperor) was established in the Korean Empire period where the last Joseon rulers were regarded as "Emperors" and various historical kings were elevated to the status of Emperor as a way to indicate the end of the kingdoms
tributary (note, not "vassal") status to the Chinese empire.
The relationship between Korean kingdoms and the Chinese dynasties were close, yes, but not always. From the Gojoseon period (approximately 2000 BCE) to the Samguk period (668 CE), Korean tribes, kingdoms, and chiefdoms were in conflict with the Chinese dynasties so much so that they were referred to collectively as part of the Dongyi, the so-called "Eastern Barbarians." Shilla recognized the superiority of the Tang dynasty in order to gain an alliance to defeat its rival and gain supremacy over the region; however, shortly after succeeding, the Shilla kingdom (57 BCE - 935 CE) turned against Tang and drove them out. Manchuria became the Balhae kingdom which was also waged war with the Tang dynasty until a compromise was reached.
That some members here consider Korea as merely a "vassal" of China indicates that people seem only to remember Korea for the ultra-conservative Joseon period and the Japanese colonial period. Believe it or not - Korea has a MUCH longer history than what short-term memories can recall.
On the note of the Japanese tenno: The Japanese unlike the Chinese regarded their tenno as a virtual divinity, descending from the Goddess of the Sun, Amaterasu. Japan for the most part kept to itself so whether or not the Japanese regarded their emperor as an equal or divine didn't really matter all that much to the mainland. From the Asuka Period onwards, they were considered equal in title, but the Japanese did not use the Chinese "Son of Heaven" title since that was reserved for the Chinese monarch. Also, Japanese emperors were typically puppets of their own subject; the regents in the Heian Period and later the Shoguns of the Kamakura, Muromachi, Azuchi-Momoyama, and Tokugawa Periods held the real power.
BTW, Civilization IV likely *isn't* that popular in Korea; none of my 422 students have the vaguest idea of what it is.
And (before anyone makes such an accusation as a weak means of discrediting anything I say here), no, I am not Korean; I am part Vietnamese and part Chinese with Min and Yue heritage. I am a pho lover and damn proud of it.
