There are a few connections we can discuss as to how the story of Enlil's flood relates to Gilgamesh and his quest.
In "The Search for Everlasting Life" a goddess (Siduri) tells Gilgamesh, "You will never find that life you for which you are looking. When the gods created man they allotted to him death, but life they retained in their own keeping."
This precept is borne out by narrative related in "The Story of the Flood" as it dramatizes one way in which the will of the gods dominates the fate of mankind. When gods like Enlil become bothered by mankind, it is well within their power to destroy man and man's world. Enlil's actions give force to this notion.
Seen in this light, we can interpret Gilgamesh's quest as, in a way, setting his own (mortal) will against that of the (immortal) gods. In his quest, Gilgamesh seeks to overturn the natural order. Although he is part god, Gilgamesh is still part of the mortal world. In seeking to escape the fate of his brother Enkidu, Gilgamesh is essentially attempting to alter the fate dictated by his nature.
Another reason Enlil's story can be seen as important is that this story shows that exceptions can be made. The natural order can be overturned! But...there are conditions that must be met.
When Utnapishtim is given a command to save mankind (in order to please the gods), he is granted eternal life. Thus we see that Gilgamesh is not on an impossible quest.
Importantly, the pleasure of the gods remains as the principally important element in deciding what the natural order will be and/or when it might be altered. Utnapishtim does as he is told. He does not achieve immortality through pride, strength or any kind of self-proclamation. He obeys.
The final word on "what will be" comes from the great god, Ea. It is Ea who chastizes Enlil and suggests that the flood was too extreme a punishment for mankind's transgressions. A sense of balance and proportion is necessary.
This wisdom from Ea raises a question about the quest of Gilgamesh: Is his quest for eternal life characterized by a sense of balance and proportion? Also, is his quest characterized by a sense of humility and respect for the will of the gods?
Going one step further, we might argue that the moral of Enlil's story serves accurately to predict the measure of success that Gilgamesh does (and does not) achieve on his quest. If proportionality is the over-arching moral, the fact that Gilgamesh (2/3 god and 1/3 man) is able to acquire a youth-giving plant (and then lose it) and is unable to achieve immortality seems to match well with the moral.
"At the end of his quest with Utnapishtim, [...] Gilgamesh's lesson is that humans, though they cannot escape their mortality, can be transformed through experience" (eNotes).