Enkidu and Gilgamesh create a dichotomy (a system of two contrasting elements) between civilization and pre-civilized peoples. Enkidu is also 2/3 animal whereas Gilgamesh is 2/3 divine. Gilgamesh is a powerful ruler, wise in the ways of society, while Enkidu is wise in animal-like ways.
The two men are thus almost total opposites. But they are also very similar. Enkidu is Gilgamesh's soulmate, but a more animal-like and innocent version of him. Many heroes in literature beat their opponent and then instead of killing them take them as a friend--King Arthur and Lancelot, Robin Hood and Little John.
Gilgamesh sends a woman to the wilderness to teach Enkidu how to be civilized, but Enkidu also "humanizes" Gilgamesh. He teaches him how to share and grow, and makes him less arrogant and lonely. The effect on Gilgamesh's life is so great that when Enkidu dies, Gilgamesh is inspired to go on a quest for immortality.
Thus the whole plot is unleashed by Gilgamesh at first being incomplete, defeating a man who is his opposite and yet his peer, befriending him and becoming whole, losing him, and then realizing the dangers of death and becoming inspired to find eternal life.
It could be read as the story of how we need to be whole to understand how precious life is.