The short answer is: not much. The long answer follows.
Biblical scholars often give more weight to a manuscript that is older, as compared to a more recent manuscript of the same passage. The reason is that copyist errors, and editing, and glosses (comments in the margins) can have an effect on the text of any particular edition. This means that a later text might have errors that an earlier text would not have.
Sometimes a copyist error is not an obvious typo. The copyist inadvertently slightly rephrases a verse, because the rephrasing sounds more natural to him, or expresses the meaning that he perceives in the passage more clearly. I say ‘inadvertently’ because often this rephrasing occurs without the person realizing it. For example, suppose you are proof-reading an article that you wrote. You might miss one or more typographical errors because, as you read the text, you mind is focused on the meaning, more than the letters and their order. Your mind reads the correct word, despite the typo. Similarly, it is also easy, when copying many verses of Scripture, one after another, to accidentally rephrase a sentence, in a way that makes sense, even in a way that improves the text. This improvement does not add or subtract or change any meaning, but merely clarifies the truth that is already present in the text. So the later text might actually have the better wording.
Sometimes the differences between earlier and later manuscripts is due to editing. The scholars who have worked with Sacred Scripture over the centuries have treated the Word of God as if its truths were found in its meaning, rather than in its letters. Sacred Scripture is not: ‘this exact set of letters, in this exact order, in this language only’. The Bible is its meaning, more than its letters. And therefore, the Bible can be translated into any language, while still preserving and expressing its truths. You do not need to know Aramaic or Hebrew in order to understand the teachings of Jesus (who mainly taught in those languages).
But editing can also improve the text of the Bible. An editor can compare different wordings, from different editions, manuscripts, languages, and then phrase the text of his edition in a way that clarifies a meaning, which would otherwise be more obscure. Of course, he can also obscure a meaning by his editing. Sometimes, the word choice of a translator or editor of the Bible obscures one level of meaning, and clarifies another; or vice versa.
It is said that some phrases or even whole verses in the Bible entered the text from glosses, from comments in the margin that later copyists mistakenly included in the text. This occurs because, when a copyist makes an error, he can’t erase it. He can’t crumple up the page and start with a fresh sheet. Parchment was very expensive. So he would write the correction in the margin. If he accidentally skipped over a phrase or verse, he would just add it in the margins. But then again, the commentary to the Bible was also placed in the margins. So occasionally, a copyist might mistake a margin gloss (commentary or explanation) as part of the verse. However, I consider that modern Biblical scholars are far too quick to remove phrases and verses from Sacred Scripture, on the allegation that it was a gloss, or on some other basis.
Another process that results in alternate phrasings for a verse is translation. Jesus taught mainly in Aramaic. So when Luke, for example, wrote his Gospel in Greek, he needed to translate from the Aramaic. When the Gospel of Luke was translated later on into Latin, many different persons made translations. As Scripture was translated into many other languages, the translation decisions had an effect on later manuscripts, because sometimes a scholar would work with several editions of a book of the Bible, in various languages. And this would have an effect on editing in any newer edition of that book.
So there are some good reasons why the readings in older manuscripts should prevail. But there are also good reasons why later manuscripts, with a different reading of the text, should prevail instead. Nothing can be added, subtracted, or changed in the truths that are revealed in Sacred Scripture. But the copying, translating, and editing of the Bible can improve the particular expression of these truths. So, over time, the text of the Bible evolves, and the truths that were always present in that text become ever clearer. The more good editions and translations of the Bible that are produced, the more clear the innumerable truths of the Bible become.
But it is not copyists, translators, and editors who have influence over the evolution of the Bible. Sacred Scripture is joined, within the Church, with Sacred Tradition and the Magisterium. The Living Tradition makes use of many different editions of the Bible, and this use vets each edition. Any errors by copyists, translators, or editors are noticed and rejected, or corrected, or set aside. The Living Tradition also continually meditates on Scripture and reveals the fruits of that meditation, from the hearts and minds of the faithful to the hearts and minds of the faithful, and this continuous revelation of understanding affects the work of Scripture scholars also.
The Living Magisterium does the same. The Magisterium continually makes use of Sacred Scripture in teaching the Faith to the faithful, and so the meanings of each passages are continually becoming more and more fully revealed. And this continuous revelation also affects the work of Scripture scholars.
This use of the Latin Bible by the Living Tradition and the Living Magisterium for many centuries was one reason why the Council of Trent gave the Latin version of Sacred Scripture a special place among Bible translations:
“But if any one receive not, as sacred and canonical, the said books entire with all their parts, as they have been used to be read in the Catholic Church, and as they are contained in the old Latin vulgate edition; and knowingly and deliberately contemn the traditions aforesaid; let him be anathema.”
So a manuscript that has been in use in the Church for many centuries, of which many copies exist attesting to that widespread and continued usage, should be given additional weight over an older manuscript that was never used in the Church.
And this brings us to the question of the Dead Sea Scrolls. The versions of some books of the Old Testament among the scrolls were only used by a certain group of Essenians, living in the desert of Qumran, apart from even the society of other Jews. They used these scrolls in their isolated community, and then the scrolls lay buried from the first century to the twentieth century. Since these editions of these Bible books were never used in the Church, and were not even in general use among the Jews, they should be given much less weight than later editions. For the later editions have been vetted by the Living Tradition and the Living Magisterium, they have been vetted and even improved by constant use in the Church. Whereas these early buried editions have not been in use.
Therefore, too, Biblical scholars should not always inexorably prefer the oldest manuscripts over the newest manuscripts.