In some Skeptical circles, it is still fashionable to make the claim that the creation account of Genesis was in some sense borrowed from the Babylonian creation account, Enuma Elish (hereafter EE; those who hold this view are hereafter "EE proponents"). I have used a turn of phrase suggesting that the argument isn't held in all Skeptical circles -- the latest fad in this regard is to attribute most of the borrowing from Egyptian sources, as Greenberg does in 101 Myths of the Bible (though he posits some Babylonian influence on stories like the Flood and Cain and Abel -- see Link 1 below).
Still, it is a good idea to run through some of the arguments. My own perception is that we would expect some similarities in EE and Genesis -- and in other creation accounts as well -- if they all derived from a common source.
Some of the differences in the accounts are basic:
- EE records "successive generations of gods and goddesses" who are subject to typical weaknesses such as hunger, thirst, and sex drive; Genesis records but one God, though He had company of unspecified nature (Gen. 1:26), with no such weaknesses.
- The EE is a creation account to some extent, but most of it is devoted to describing a battle between the god Marduk (the "creator" as such) and Tiamat the goddess (who ends up being the raw material of creation), and to other non-creation issues, so that after tally, only about a third of it is on the subject of creation.
- EE played a political and cultic role in the Babylonian religion and explained Marduk's rise to chief god of Babylon; Genesis does not mention Israel, Jerusalem, or the Temple, and served no cultic function [Sarna, Understanding Genesis, 9; I would suggest that this points to the Genesis account being more original].
But, let us move to detail. Our foundational source for this essay is Alexander Heidel's classic work, The Babylonian Genesis. (U. of Chicago Press, 1942) We will address relevant points in outline form, following the order of Genesis as required.
Gen. 1:1 In the beginning God created the heaven and the earth.
All stories must start somewhere, and the ways in which Genesis differs from EE at the very beginning is quite significant. Genesis starts us "in the beginning", at a time-point which suggests nothing before.
But EE and other Babylonian creation accounts start with words like, "on the day that" or "when" -- which do not specify a beginning. The Hebrew word here means "at the first" (Numbers 15:20 "Ye shall offer up a cake of the first of your dough for an heave offering..."); the matching Hebrew word for the Babylonian record is not what is used. This feature "finds no parallels in the cosmogonies" of Babylon.
1:2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
EE also supposes a watery chaos in place, and this is where EE proponents had their biggest party. The word for "deep" here is tehowm, and EE proponents leapt upon the similarity of this word to the name of the Babylonia goddess Tiamat.
In the EE, Tiamat was the water-goddess who was slain by Marduk and used to make the watery chaos. It was supposed that tehom was linguistically derived from Tiamat, thus proving borrowing.
Substantial differences render this unlikely. Tiamat was only one of two water-deities involved in this story; the other was the water-god Apsu. Tiamat was salty water; Apsu was fresh water. Apsu, at any rate, has no parallel in Genesis at all, and the tehom is inanimate.
Of more import, the linguistic connection supposed by the critics could, if anything, only have gone in the opposite direction. The words are indeed from the same root (as are indeed, as Heidel points out, the German word for "blessed" (selig) and the English word "silly"), but Heidel demonstrates at length that for tehowm to be derived from Tiamat is "grammatically impossible" based on the rules of Hebrew as we know them. The Hebrew tehowm has a masculine ending; Tiamat is feminine.
A loan word from Babylonian to Hebrew would retain the feminine; we would not expect tehowm but tiama or teama. Hebrew would also not add the H unless it were found in the original word (i.e., it would have to have been Tihamat). Heidel's conclusion is that the two words probably go back to "a common Semitic form," rather than that one was derived from the other .
I can mention an observation of my own here. If Genesis was an effort to "clean up" the Babylonian myth for Hebrews, it wouldn't make a lot of sense to use a word with such a clear linguistic connection to the name of a Babylonian goddess. Genesis could simply have referred to the "waters" as it does later on.
It is also worth mentioning a special connection that was made by EE proponents who suggested that the EE represented a symbolic form of the rainy season and flood cycle and Babylon, and that this was proof of borrowing by the Hebrews, because they kept this form in spite of living in arid Judea. Heidel responds by noting that the EE proponents hadn't studied Babylonian climate very well -- the rainy season and the flood season there come at entirely different times of the year; if indeed there is a "rainy season" as the area gets only about 6 inches of precipitation a year. 
1:3-5 And God said, Let there be light: and there was light. And God saw the light, that it was good: and God divided the light from the darkness. And God called the light Day, and the darkness he called Night...
Like EE, Genesis says that light was around even before the creation of the luminaries. A difference here is that in EE, the light was an attribute of deity, whereas here, it is a creation of the deity.
1:6-10 And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters. And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament: and it was so. And God called the firmament Heaven. And the evening and the morning were the second day. And God said, Let the waters under the heaven be gathered together unto one place, and let the dry land appear: and it was so. And God called the dry land Earth; and the gathering together of the waters called he Seas: and God saw that it was good.
The dividing of substances is a commonality in several creation accounts across cultures -- it is found in Egypt, Phoenicia, and India as well, though the elements in question may differ (water, an egg, etc.)[114-115]. The point being, that it makes better sense to postulate an ancient common source than to suggest borrowing.
1:11 And God said, Let the earth bring forth grass, the herb yielding seed, and the fruit tree yielding fruit after his kind, whose seed is in itself, upon the earth: and it was so....(1:19-25)
This, and the creation of animals, has no parallel known in the EE. It is possible that the creation of vegetation and animals was recorded on a part of the EE that was lost; Heidel reports a gap in the record, but doesn't suppose that there was much room for such a report, which would have to fall between astronomical data and a plea from the gods .
I would suggest that it is easier to imagine the Babylonians dropping this element rather than Genesis adding it.
1:14-18 And God said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days, and years: And let them be for lights in the firmament of the heaven to give light upon the earth: and it was so. And God made two great lights; the greater light to rule the day, and the lesser light to rule the night: he made the stars also. And God set them in the firmament of the heaven to give light upon the earth, And to rule over the day and over the night, and to divide the light from the darkness: and God saw that it was good.
Both EE and Genesis record the creation of the luminaries, and in both cases say that they were for light and time-keeping purposes -- no surprise, since all cultures worldwide use the luminaries for the same purpose. The Babylonian account does take a more "astrological" view, however, as it makes a point of the creation of the zodiac and the partitioning of constellations within.
Genesis also reports the creation of luminaries in the reverse order of the EE (which lists the stars first). Genesis is also missing EE's reference to gates at the east and the west of the sky through which the sun and moon pass .
One is constrained to ask how critics, who suppose the Hebrews to believe in a solid sky and a flat earth (see link 2 below), think that this little tidbit was left out of a "copycat" story.
1:26-30 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth. And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat. And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.
With this we come to perhaps the most significant difference between EE and Genesis. The creation of man at first blush seems very similar -- God creates man from dust, and imputes with life with His breath or spirit; Marduk gives Ea (another god) a plan to create man, leading Ea to go and kill Kingu (another god yet) and mix his blood with dirt to make man.
The basic principles of dirt and divine substance appears in both accounts, but few would deny that Genesis is more sophisticated -- leaving us again to wonder whether positing a "clean up" job is more plausible.
A larger difference, however, emerges in terms of man's purpose. In the EE, man is created because Marduk was prompted in his heart to "create ingenious things". Once that is done, man's purpose is to serve the gods, build their temples, and make sacrifices to them. Men are, essentially, the gods' boot-polishers.
But in Genesis, man is not a servant to God; he is God's agent. For more on this, particularly image-language, see Chapter 1 of my book, The Mormon Defenders (link 3 below). There is great significance in the point that in other societies than Israel, the "image" language is applied only to rulers. Socially it is more likely that this designation was taken from all men by power-hungry rulers than that it was expanded by the Hebrews to include all men.
Genesis 2:2-3 And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made. And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.
The gods of the EE "rested" also, after a fashion -- they threw a big party, one that takes up almost two tablets of the EE out of seven, in which Marduk is honored. In that context it is worth noting that some saw a connection between Genesis' 7 days and the EE's 7 tablets, but as noted, the creation takes up only four of EE's tablets, and the EE does not lay out a seven-day pattern of creation.
Our summary conclusion: The views of EE proponents simply do not correspond with the data -- and thus it is not surprising that most borrowing-proponents have sought their parallels elsewhere. (For more on those other stories, see the series here by the Christian ThinkTank.)