Ok, many, many things...

It is not so much that the Dead Sea Scrolls do not get more attention at this point, it is more the fact that the attention they do receive in the media seems to be more sensationalist than productive, accurate, or helpful. This is particularly the case in examples like the recent "new finds" of scrolls which were in the media not too long ago.

What I will first do is name several sources and scholars whom you should consult and then I will provide more specific information as pertains to the content of the scrolls and the impact that they have had on biblical scholarship in the long-term.

First, the scholars: I would point you to the following individuals: Frank Moore Cross, Elisha Qimron, Geza Vermes, Jodi Magness, Lawrence Schiffman, John J. Collins, and James van der Kam. These people should provide you with more than enough information on the background, findings, interpretations, etc. of the Dead Sea Scrolls.

As I continue: one assumption I will be making in this post is that the community at Qumran produced the majority of the texts found in the caves surrounding the site. It is very likely that some texts were BROUGHT to the site from outside, but I am convinced that many, if not most or all, were composed on site. (They did, after all, find many more ink wells in ruins than they have anywhere else.)

As far as their impact on biblical studies is concerned, what the Dead Sea Scrolls provided us with was a wealth of knowledge about the nature of the development of many biblical texts. For example, we might readily consider the Greek version of Jeremiah vis-à-vis the Hebrew version of Jeremiah. The Greek version as attested at Qumran turned out to be about 5/6 the length of the Hebrew version and it was also rearranged in an entirely different order. This suggests to us that ancient tradents of what would eventually become biblical texts were comfortable with a pluriformity of textual traditions. That is to say, they were comfortable with having various editions of the same text in the same corpus. This also has implications on how we understand the development of the text as well as the development of the canon of the Bible. Secondly, we might readily consider the fact that there is no copy of Esther attested at Qumran nor is there a copy of Nehemiah. This very well may be the case because of the fact that Esther does not mention god once in the entire book. Nehemiah may not be attested by the Qumran scrolls because of the fact that so much of it is parallel to Ezra. There is another comment in this thread that suggests that Daniel was not found Qumran; However, this is not the case. There are several documents found Qumran (e.g., 4QPrayerNab) that attest to Daniel being preserved by the Qumran community.

One of my favorite examples about the impact of the Dead Sea Scrolls on Biblical studies is Deut 32:8-9. 4QDeut^q attests a reading different from the Masoretic text of the Hebrew Bible. The Masoretic text of the Hebrew Bible is as follows:

בְּהַנְחֵל עֶלְיוֹן גּוֹיִם בְּהַפְרִידוֹ בְּנֵי אָדָם יַצֵּב גְּבֻלֹת עַמִּים לְמִסְפַּר בְּנֵי יִשְׂרָאֵל׃ כִּי חֵלֶק יְהוָֹה עַמּוֹ יַעֲקֹב חֶבֶל נַחֲלָתוֹ׃

My own translation:

When Elyon apportioned the nations, when he divided up mankind, he situated the borders of the peoples according to the number of the Israelites, because the portion of Yahweh is his people, the portion of Jacob his allotment.

Notice that I have bolded "Israelites." Where the Masoretic text has "Israelites," 4QDeut^q attests "the sons of the gods" (בני האלהים). This is a significant variance from the Masoretic tradition. What this tells us is that Yahweh was at one point not the chief deity in the Canaanite pantheon. Rather, Elyon (= "the most high") was the chief deity Yahweh was a deity subordinate to Elyon. At some point, the Masoretic textual tradition altered what we have maintained in 4QDeut^q to provide a more theologically palatable reading. "The sons of the gods" can only be understood as Junior deities. The phrase appears in Semitic with only this understanding and none other.

Furthermore, many of the noncanonical texts found at Qumran are what we call sectarian documents. These documents lay out many of the rules and regulations that were maintained for the community that supposedly lived at Qumran. For example, we may very well consider the document known as the "community rule," which laid out stipulations for how individuals were to enter and join the community and how they were to live as a part of that community. These sectarian documents were influenced and informed by other sacred texts, although in many places they depart from those sacred texts in favor of their own interpretations or understandings of their own traditions.

We also gain greater insight into the way in which this community interpreted the sacred texts they had at their disposal. We very well may consider Pesher Habakkuk as an example of this kind of text. In Pesher Habakkuk, we have the combination of a biblical texts with significant amounts of commentary that pertains to the community that produced the text. Here, we see mentions of the "teacher of righteousness," "the sons of light," and "the sons of darkness." Presumably, the teacher of righteousness was the leader of the Qumran community, the sons of light were those who have been accepted into the community, and the sons of darkness with those who stood opposed ideologically and/or theologically to the Qumran community.

EDIT: I made this post with voice to text recognition and didn't proof read it. Fixing things now...

They do get tons of attention! Most modern bible translations use them to an extent, especially to get an extra textual reference where the Masoretic text (the standard Hebrew) and the LXX (the Septuagint, an ancient Greek translation of the then-current Hebrew) disagree.

In terms of "matching what we were taught", that's the wrong way of looking at it IMO. They're biblical manuscripts. They can illuminate what things were scribal errors, what might've been censorship. They don't have some sort of extra theology.

There are, however, some sectarian texts, that let us know what the Qumran community believed. While it's an interesting topic, the people there seem to have been a minority sect. It tells us what some people in ancient Judea believed, not what ancient Judaism was actually like in total. So they don't match what anyone's teaching, because they're from a long-extinct sect.

As for whether any of this has influenced modern theology, not really. The things that'd be important are scribal differences in biblical texts that theologies are based on. While I've heard arguments that the DSS disprove certain theologies as scribal errors in the standard Hebrew texts, all those I've seen are polemic arguments by people who don't know much about textual history (doing things like drawing conclusions by comparing translations of manuscripts, not manuscripts themselves). The only real one is that it's evidence against the modern biblical text being exactly transmitted down, but that was pretty clear beforehand, too.

However, practically speaking people aren't going to start printing bibles based only off the DSS and translations of it. While there are copies of every book of the Hebrew canon except Esther, most of them are fragmentary. We might have a few lines, a chapter or two, and that's it. It's enough to reference it usefully when translating, but not enough to replace the Masoretic Text, which is the standard Hebrew one.

The DSS contain the earliest versions of almost every book in the Hebrew Bible (I believe Daniel is the major exception). Religions don't function like science, they don't always readily accept new information as it becomes available, but their main significance to religion and apologetics is that it shows the Hebrew Bible was largely handed down free from gross error, mistranslation, and later editing. But this doesn't really prove anything new, scholars already knew that Near Eastern societies were capable of maintaining an accurate textual tradition, and we had already had other sources to look into this. It may be a good response to people saying "oh it's a translation of a translation who knows what It originally said...", but that's not a position scholarship has really held for the Old Testament. Critical study has been more concerned with how and when these documents were actually created than their transmission. For what it's worth, these studies typically do not side with the traditional dating and authorship claims.

DSS also contains some interesting texts on esoteric, mystical Jewish religion and the way the community that produced the scrolls actually lived. This may have some implications for how religion evolved in the inter-testamental period, but nothing that really confirms or refutes any major claims, as far as I am aware.

I would highly recommend Robert M. Price's podcast The Bible Geek, as well as his many books including The Pre-Nicene New Testament and The Incredible Shrinking Son of Man. He often talks about various things found in the Dead Sea Scrolls, along with many other non-canonical Biblical texts.