The Dead Sea Scrolls contain the earliest known copy of the Hebrew Bible (which Christians refer to as the Old Testament). Before the discovery of the Dead Sea Scrolls, the earliest copy of the Old Testament written in Hebrew were the Masoretic texts (c. 1000 AD). The Dead Sea Scrolls predate the Masoretic texts by over a thousand years, with some of the scrolls dating to as early as 100 BC.
Although there are copies of the Hebrew Bible known as the Septuagint dated to as early as the 4th century, this was written in Greek, not Hebrew. The King James Version of the Bible is based on the 10th century Masoretic texts since it was thought that the Septuagint mistranslated the original language in several places.
The Dead Sea Scrolls contain the entire book of Isaiah as well as parts of every canonical book in the Hebrew Bible except for Esther, along with several Apocryphal writings. While there are numerous spelling and grammatical differences, the Dead Sea Scrolls are essentially the same as the Masoretic texts for the most part. However, there are a few major differences.
For example, Psalm 145 is an acrostic poem. The first verse begins with aleph, the second verse begins with beth, and so forth all the way through the Hebrew alphabet. However, the Masoretic text is missing the verse that should begin with nun. This lost verse was found again in the Dead Sea Scrolls.
Growing up as a Mormon, I heard the claim that the Dead Sea Scrolls proved that Joseph Smith's Translation of the Bible was correct and that the King James Version of the Bible was translated incorrectly. There's no question that the KJV contains numerous translation errors, however, do the Dead Sea Scrolls validate Joseph Smith's Translation?
The short answer is no. Joseph Smith's version of Psalm 145 is exactly the same as the King James Version (specifically Blayney's 1769 revision of the KJV). He didn't notice that a verse was missing.
The Dead Sea Scrolls give us an extra verse before verse 1 of 1 Samuel 11, which I will call "verse 0":
Nahash king of the Ammonites oppressed the Gadites and the Reubenites viciously. He put out the right eye of all of them and brought fear and trembling on Israel. Not one of the Israelites in the region beyond the Jordan remained whose right eye Nahash king of the Ammonites did not put out, except seven thousand men who escaped from the Ammonites and went to Jabesh-gilea" (1 Samuel 11:0)
This verse is missing from both the King James Version and the Joseph Smith Translation. Reading the following passage in the KJV or the JST makes it seems like Samuel ripped his own mantle:
And as Samuel turned about to go away, he laid hold upon the skirt of his mantle, and it rent. (1 Samuel 15:27, KJV & JST)
However, the Dead Sea Scrolls reveal that it was actually Saul who did the ripping:
And as Samuel turned about to go away, Saul laid hold upon the skirt of his mantle, and it rent. (1 Samuel 15:27, DSS)
In 1 Samuel 17:4, the DSS give the height of Goliath as "four cubits and a span" which is over 7 feet tall. The Masoretic text, which was written a millennium later, gives his height as "six cubits and a span" which would make him over 9 feet tall. It's only natural that some stories grow with the telling. Again, the JST is exactly the same as the Masoretic-based KJV.
As an aside for fans of Antiquities of the Jews, (Go Team Josephus!), Josephus' account of first and second Samuel matches the Dead Sea Scrolls version more closely than either the Masoretic text (MT) or the Septuagint (which is commonly abbreviated as LXX for some reason). This provides some evidence that the writings of Josephus are indeed based on an earlier version of the Hebrew Bible than we currently possess.
Next, let's go back to where it all began:
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. (Genesis 1:9 KJV & JST)
The Dead Sea Scrolls add an extra sentence:
And God said, let the waters underneath the heavens be gathered together in one gathering, and let the dry land appear. And it was so. And the waters under the heavens gathered to their gatherings and the dry land appeared. (Genesis 1:9 DSS)
Joseph Smith didn't get this one right either, however the LXX does give the longer version of the verse, making it more accurate than the Masoretic in this case. Another instance where the LXX matches the DSS, but the Masoretic does not is Psalm 22:16 where the DSS has "they have pierced my hands and feet" while the Masoretic has "like a lion are my hands and feet" which is just crazy talk.
Another example of this can be found in Deuteronomy 32:8 where both the DSS and the LXX have "according to the number of the sons of God" while the Masoretic (and therefor the KJV and JST) have it as "according to the number of the sons of Israel" which actually doesn't make sense when put into context:
When the Most High divided to the nations their inheritance, when he separated the sons of Adam, he set the bounds of the people according to the number of the children of Israel. (Deuteronomy 32:8)
This verse, as the KJV and JST have it, tells us that all the nations of the world are children of Israel. The original Dead Sea Scrolls version implies the older Jewish theology that the world was divided up into different nations each with its own god who was the son of the most high God.
The paragraph where Joshua constructs an altar is located at Joshua 8:30-35 in the KJV and JST. The LXX puts the paragraph a little bit later (Joshua 9:7-8), but in the DSS, the paragraph is located at the beginning of Joshua 5. The only source that agrees with the Dead Sea Scrolls in this case is Josephus. (Go Josephus!)
Exodus 1:3 has "Joseph and Benjamin" in the DSS, but the KJV and JST both leave out Joseph. Judges 6:7-10 are not found in the DSS and are therefor a later insertion, but again the JST blindly follows the KJV.
The DSS version of Psalms is radically different from the version we have today with the psalms appearing in a different order, some psalms missing, and 15 new psalms. The JST actually differs from the KJV when it comes to Psalm 32:
Blessed is he whose transgression is forgiven, whose sin is covered. (Psalm 32:1 KJV)
Blessed are they whose transgressions are forgiven, and who have no sins to be covered. (Psalm 32:1 JST)
Unfortunately, it doesn't match with the DSS version, because Psalm 32 is missing from the DSS entirely, as are psalms 70 and 90 (Joseph Smith copies the KJV exactly for these two). Joseph Smith does completely rewrite Psalms 11, 14, and 24, unfortunately, the DSS versions of these psalms are basically the same as the KJV.
There are a few minor differences between the DSS and the Masoretic text in Psalm 138:
To David, I will praise you with all my heart, before God I will make your music. (Psalm 138:1 Masoretic text)
To David, I will praise Yahweh with all my heart, before Yahweh God I will make your music. (Psalm 138:1 DSS)
If I walk in among trouble you will revive me, over the anger of my enemies you will send your hand you will rescue me, your right hand. (Psalm 138:7 Masoretic text)
If I walk in the middle of trouble you will revive me, over the anger of my enemies you will send your hand you will rescue me, your right hand. (Psalm 138:7 DSS)
Joseph Smith didn't notice these small differences, but he does rewrite the next verse.
Yahweh will end on behalf of me, Yahweh, your mercy is for eternity, the works of your hands will not sink. (Psalm 138:8, the same in both Masoretic and DSS)
The Lord will perfect me in knowledge, concerning his kingdom. I will praise thee O Lord, for ever; for thou art merciful, and wilt not forsake the works of thine own hands. (Psalm 138:8 JST)
Perhaps the most major difference between the JST and the KJV is that the JST completely removes the Song of Solomon. However, it is included in the DSS. The DSS lack Esther, however the JST includes it.
At the risk of beating a dead horse, I'll give a few more examples from the book of Isaiah. The Book of Mormon quotes from Isaiah quite extensively so it was obviously a very important book to Joseph Smith.
Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel. (Isaiah 7:14 KJV & JST, also 2 Nephi 17:14)
You can't tell from the KJV, but in the Masoretic text, the final verb in this verse is "she will call" indicating that the virgin (or young woman as it's more correctly translated) is the one who names the son. In the DSS, the verb "he will call" is used instead.
For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace. (Isaiah 9:6 KJV & JST, also 2 Nephi 19:6)
Instead of "The mighty God, The everlasting Father, the Prince of Peace" the DSS has "Elgibor, the father of Ad, ruler of Hashalom." The reason this is so different is due to minor manuscript differences. The name Elgibor in the DSS gets changed to "mighty God" in the Masoretic text simply by inserting a space ("el gibor" means "God is a warrior"). Similar minor differences change "father of Ad" into "everlasting Father" and "ruler of Hashalom" into "ruler of peace".
The Masoretic text translates Isaiah 53:11 as "from the labor of his soul, he will see, he will be satisfied". The Dead Sea Scroll text translates to "from the labor of his soul, he will see light and he will be satisfied". The JST again follows the KJV.