Enkidu's "fall" does share similarities with the expulsion of Adam and Eve from the Garden of Eden, though his is not so much a case of sin as it is a case of civilization.
Enkidu starts life as a wild man, only communing with animals and living as one with the natural world. After the sacred prostitute Shamhat sleeps with Enkidu, he has lost his "wild" state. He is shunned by the animals, who see him no longer as one of their own but as a man. Enkidu finds this state of affairs initially sorrowful, and in his low moments, he blames Shamhat for bringing him into the human world. But without this loss of animal communication, Enkidu would have never found love with Gilgamesh. His "fall" was necessary to come to greater understanding.
In traditional Christian doctrine, the fall of Adam and Eve is viewed as wholly tragic. Adam and Eve are more civilized than Enkidu in that they are already ordained by God as being higher than the animals; however, they do commune with the animals, and they exist in a state of innocence, not even aware of their own nudity. When the two eat of the fruit of the Tree of the Knowledge of Good and Evil, then add insult to injury by trying to shirk their responsibility in the act before God, they are thrust into a fallen state and expelled from Eden.
However, there are alternate views of the Adam and Eve story, from both Christians and Jews, which line up more with the idea of Enkidu's "fall." Rabbi Harold Kushner argues in his book When Bad Things Happen to Good People that the fall is more an allegory about what it means to be human than a tale of why man is corrupt and deserving of suffering. While the animals remain unaware of morality, Adam and Eve, in eating of the Tree of the Knowledge of Good and Evil, become aware of morality and therefore suffer in making moral choices. In Kushner's view, this is what makes humans different from the animals, just as Enkidu's carnal knowledge of a civilized woman (who is also a representative of the gods in her profession) makes him closer to gods than to beasts.
As for comparing The Epic of Gilgamesh to the Bible, there are a few reasons why scholars do so. Firstly, Gilgamesh features one of the earliest Great Flood narratives, even predating the one presented in the Book of Genesis. Secondly, both are early works of literature which detail how earlier civilizations related to the divine and to the never-dated question of what makes us human.