Both Genesis and the Gilgamesh Epic contain accounts of?

The Flood

Noah and Utnapishtim

All the above. The Old Testament came almost entirely from this old fable.

Genesis came first

Copying is flattery

...the flood.

I an really having a hard time understanding how people say the epic of Gilgamesh is the inspiration of the Biblical Flood account

THe differences between Flood accounts from the Bible and ancient Babylon, P. J. Wiseman wrote in New Discoveries in Babylonia About Genesis: “The Bible account is simple in its ideas, and irreproachable in its teaching about God, while the Babylonian tablets are complex and polytheistic. The difference may be compared to that between the pure waters of the springs at the source of the Thames, and the contaminated waters of the docks of London. There are resemblances between a river at its source and at its termination, both are in one sense the same river; so in Genesis we find the story at its pure source, while in the Babylonian it is seen at its contaminated development.”

As for the Bible’s being dependent upon Flood accounts from ancient Babylonia, the Lexikon zur Bibel by Fritz Rienecker contains the remark: “A literary dependence of the Biblical, entirely unmythological Flood account on the Babylonian stories appears to be, however, totally unlikely in view of the differences of both texts in manner and contents.”

From study of Bible chronology, the writer of Genesis did not need to draw upon any Babylonian legend. Because of the overlapping of life-spans, the truth about the Flood could easily have been handed down by Noah’s son Shem (who was an eyewitness) through just three human links to Moses, the writer of Genesis. It is unreasonable to think that the Hebrews, who worshiped the same God as Noah did, would not have included an event of such importance in their history.

Other Bible writers had endorsed the Genesis account. For example, Isaiah and Ezekiel called attention to Noah and the Flood. (Isa. 54:9; Ezek. 14:14, 18, 20) The apostles Peter and Paul made specific references to the Flood. (1 Pet. 3:20; 2 Pet. 2:5; 3:5, 6; Heb. 11:7) And all such Bible writers, including Moses, were “inspired of God,” which gives me assurance as to the truthfulness of their accounts.—2 Tim. 3:16.

Jesus Christ, too, acknowledged that the Genesis account was the truth. When speaking of the coming destruction of the present wicked system of things, he said: “For as they were in those days before the flood, eating and drinking, men marrying and women being given in marriage, until the day that Noah entered into the ark; and they took no note until the flood came and swept them all away,” so it would be at this system’s end.—Matt. 24:37-39.

The flood.

Many cultures from that part of the world have some legend of a great flood, indicating that at some point one probably did occur.

I'm pretty sure Gilgamesh predates both Moses and Babylon. And of course it wouldn't share the same cast of characters as Jewish legend.

fun fact: Gilgamesh was 2/3 divine. Consider that mathematically for a moment.

Well the answer, surprisingly, is both the "fall of Adam" and the Flood story. The story of Shamhat's seduction of Enkidu has multiple points of contact with the story of the garden of Eden. The main difference is that the "civilization" of Enkidu is presented in a very positive way, whereas the "fall" of Adam and Eve is described in terms of loss and punishment. This may reflect different attitudes to sexuality and to the attractions of a very powerful and sophisticated mesopotamian culture on the part of the biblical writer.

As for suggestions that the biblical flood story must have come first or that it is "purer", virtually all scholars of ancient near eastern literature agree to the priority of Gilgamesh. And no, the "standard" version of the epic was not being modified in 500BCE. The so called "standard" version took its final form around 1300 to 1000BCE and scholars believe that this is when the long version of the flood story was incorporated into the epic. However, the flood story in Gilgamesh is based almost word for word on the still older flood story found in the epic of Atrahasis. This earliest copy of Atrahasis dates to 1700-1800BCE. So the Mesopotamian story is definitely older than even the most conservative dating of the book of Genesis (though most suggest that Genesis in its current form was written around 600-500BCE - the time of the Babylonian captivity). Furthermore, any suggestions about oral prehistories of either story must remain completely in the realm of conjecture: in literary terms the biblical story was not the first.

There are also several good indications that it was the mesopotamian flood story that influenced Genesis and not the other way around. Firstly, greater societies generally influence lesser societies. For example, I suspect that people in many countries around the world would be at least partly familiar with north american history (the pilgrim fathers, the civil war, the battle for the plains, etc) due to the enormous influence of American popular culture. On the other hand I wonder how many US citizens are aware of the history and mythology of Scotland, or the Congo, Australia, Thailand or Spain. Secondly, Mesopotamia (literally the "land between two rivers") was prone to constant flooding whereas Canaan was not. Thirdly, the only geographical landmark mentioned in Genesis is the Ararat mountains in norther mesopotamia. Finally, the epic of Gilgamesh was a widespread "classic" in the ancient world due to the fact that it was used as a standard text for students of cuneiform. Fragments have even been found in northern Israel. There is no such evidence for the influence of the book of Genesis from this period.

Additionally, it is not enough to say that many cultures have flood stories so that any similarities between Gilgamesh and Genesis are inconsequential. It is well documented that no other flood story has such close parallels to Genesis as that recorded in the epic of Gilgamesh. They use exactly the same basic framework of events, with some very minor differences in detail, in exactly the same order. The main difference between the two is theological. One is monotheistic, the other polytheistic. Genesis introduces the idea of a covenant or contract between God and humankind that is absent in Gilgamesh. Christians like to think that the biblical version is more moral since God sends the flood due to human wickedness and violence. Personally I rather like the Babylonian version where the flood is seen for what it really is - mass murder. In this version the gods lament the overwhelming loss of life and the god who is the main instigator of the flood is condemned for his actions. On the other hand Utnapishtim is given immortality because he has preserved life.

Incidentally, although flooding of the cities of Mesopotamia was not infrequent, there is no historical or archeological evidence for a flood that affected the whole region at the same time. It is usually surmised that, in typical middle eastern hyperbolic fashion, the experiences of local floods were exaggerated or extrapolated back to a primeval universal flood myth. For discerning readers, the lessons of this ancient myth are in no way dependent upon its historicity and can, in fact, be completely missed by those who like to wave the banners of biblical inerrancy and construct fantastically misguided theme parks.

There are about 300 accounts of the Global Flood around the world and this is just what you would expect.

Gilamesh account did predate the written Hebrew account in the scriptures but that does not mean it is the original account. You can clearly tell which one is the original (the Hebrew eyewitness account) because the other accounts including Gilamesh embellished theirs with added information from a later time.

It gives it's account of a huge Flood just as other cultures around the earth have there flood story. I recall reading one time that there is not a single place on earth where a group or tribe of people did not have there story of a flood that there elders or story tellers did not tell about.

the flood....

although the flood narrative in Gilgamesh is early, there's some evidence showing that other parts of the story of gilgamesh were added to relatively recently (as late as 500 BC).