This tension drives the story, and defines Gilgamesh's character. At the beginning of the poem, he is described as the greatest king, yet he is a tyrant, sending men off to war & taking women against their will. The divine in his personality creates a belief that he is above all other people, able to do what he wants with no regard for the consequences. It is because of this tension between his mortal and supernatural natures that Enkidu is created.
Enkidu is intended to be the equal of Gilgamesh, but his opposite as well. Enkidu has the wild nature, rather than the divine. It is this friendship that intends to tame them both, but leads to tragedy. Because of Gilgamesh's desire to be remembered, they seek the monster of the forest. In defeating him, Enkidu is sentenced to death by the gods. It is this death that awakens Gilgamesh to his own mortality. Although he is part-god, he is not immortal. Thus, he begins his journey to seek eternal life.