How does the bond between Gilgamesh and Enkidu affect the story in The Epic of Gilgamesh?

After defeating Enkidu in a wrestling contest, Gilgamesh and Enkidu become very close friends and companions (theirs is the dominant relationship which runs across the story). Do not overlook that most of Gilgamesh's early adventures and triumphs are accomplished with Enkidu at his side. Together they take on Humbaba, and it is Enkidu who convinces Gilgamesh to kill their opponent, when he has appealed to them for mercy. They'll also later defeat Ishtar and the Bull of Heaven together, after which Enkidu will die.
Enkidu's death represents the critical turning point in Gilgamesh's story, after which point Gilgamesh will become consumed by grief and stricken with fear of death. This proves critical in motivating him to pursue his quest for eternal life, as he travels in search of Utnapishtim. (Of course, this quest proves fruitless, for not even Gilgamesh can overcome death.)
To conclude, their relationship is deeply important (it is the most important relationship Gilgamesh has within the story), and it plays a critical role in shaping the course of Gilgamesh's life.