Q. Are the Dead Sea Scrolls the same as the apocryphal books missing from the King James Bible? If not, what are the differences between the two?
The Dead Sea Scrolls are not the same thing as the Apocrypha.
The Apocrypha is a group of books that were written in Greek within the Jewish community in the centuries before Christ. Those books are distinct from the Old Testament because they were written in Greek, not Hebrew, and they are distinct from the New Testament because they were written before Christ came, not after. So there is something about them that sets them apart as different from the books that all Christians accept as inspired Scripture. As a result, Christians hold differing beliefs about how authoritative they are. All Christians agree that they are valuable and edifying to read. But not all Christians consider them to be the inspired word of God. For a fuller discussion of this, please see this post.
The term Dead Sea Scrolls refers to a specific set of manuscripts (that is, handwritten copies of certain works) found in an area near the Dead Sea. Many of these are manuscripts of books that are in the Old Testament. Others are manuscripts of books that are part of the Apocrypha. Still other manuscripts are of books that only a few communities of Christians accept as Scriptural. (I discuss several of those books in this post.) There are also a few manuscripts relating to the specific beliefs and practices of the community whose members created these handwritten manuscripts.
So there is something of an overlap between the Dead Sea Scrolls and the Apocrypha, in the sense that some of the Dead Sea Scrolls are manuscripts of books in the Apocrypha. But the Apocrypha is basically a group of books that exists in very many copies, while the Dead Sea Scrolls are one set of copies of certain books, some of which are books of the Apocrypha.
Incidentally, when the King James Bible was first published in 1611, it did contain the Apocrypha because it was produced by the Church of England, which accepted the Apocrypha as Scriptural. But as other communities have reprinted the King James Bible over the centuries, many of them have left out the Apocrypha because they do not consider it Scriptural.
I hope this information is helpful.