To make a short: in future both institutions will become obsolete. Sometimes they are called obsolete today, but that is perhaps too pessimistic. But what was happen, that the revolution was so big? To answer this we must go back into the history and define what a library and a university was in former times.
Let us image a time, in which both were useful and an important part of society. This can be dated back into 1980s before the invention of mainstream internet. In that time, the first home computers for example the IBM 286'er were available but they capabilities were limited and it wasn't possible to use them for data interchange. Other mediums for knowledge transportation were important in that area, mainly the book, printed journals and oral presentation of a professor. The library and the university are grouped around these media. A library is a place, in which the medium book is stored, while a university is a building in which professors are speaking to their students.
Before the invention of the internet, the library and the university building were the only place for doing so. They had a monopoly on knowledge distribution. Now let us compare the situation with todays technology. If in the year 2018 somebody want's to get access to academic papers he can ask Google Scholar, if he want's to hear a professor he can watch a video ot OpenCourseware. Like in the 1980's he can also take a normal book from the library and he can also join a real university, but it is no longer a must have. In most cases, it is bad choice in doing so, because libraries are far away, and join the real stanford university is too expensive.
But if universities and libraries will disappear, what is the new modern environment in which knowledge is distributed? Classical institutions are not away, they were replaced by something which works better. That is a online repository for storing paper in digital format, that is an academic search engine, a video plattform for hosting lectures, that is computer hardware which allows the playback (smartphone and notebook) and that are product which is sold by AT&T namely a broadband internet connection. That means, in future money will be spend and time will be invested in bringing Academia forward, but the difference is, that the recipients are different. A modern form of education for example can be described by the availability of fiber optic internet connection. Building such infrastructure costs billion of US$. The idea is, instead spending money for building a university, the money is spend for optical switches and datacenters. Instead of spending 30 US$ per month for buying books the students will spend the same amount for money for the internet connection.
There is no need to discuss university and education at the same time. Instead, higher education is based on media technology. Without a medium it is not possible to transport knowledge. A medium can be: a book, newspaper, video, website, pdf file, human teacher and so on. Some of the media are equal to classical institutions for example the book and a human professor. Others like a video recording of a lecture or a pdf file are modern media which can be created and transported at lower costs. A discussion about the future role of a university in the internet makes no sense. From a technical point of view, it might be interesting for creating a website for a university, but that effort is ignoring the real capabilities of the internet. Using the internet right means to upload fulltext to an online repository and uploading a lecture to youtube. After doing so, the university needs no longer a website, because the internet is the university.
Even today, not minor technical problem makes it difficult for realizing such goal. For example, a broadband internet access is not the standard worldwide, and the amount of content available online is very low. But that a only detail problems, the general pathway is, that with more invested energy these problems can be solved. Classical universities and libraries will not simply disappear, they will be transformed into a museum. That means, a museum is build around the former building to explain future generation what a university was. This museum is not always physical visible, it will at first only be recognized as a changed debate about the role of university in societies. That means, we will see some kind of debate between people who are believing that a university is useful and other who want to describe the institution as obsolete.
The best example for such a debate are the Open Access discussion from the year 2012-2018. In most cases the idea is, how to develop existing universities forward and make them internet ready. The debate can be seen as a defend against declaring the university as obsolete. Most Open Access advocates are motivated to describe the situation optimistic, the opinion is, that universities and libraries are in general the right idea for future higher-education. But Open Access advocates lost the debate, because they are ignoring the current situation. What they are describing is not the reality, they are describing dreams and visions. The reality is, that current libraries are grouped around printed journals, while in the debate the talk is about electronic journals which are not there. Such gap in reality perception can be easily exploited by a counter movement who want's to describe universities and libraries as obsolete and are trying to replace the idea in general.
The basic idea behind a library and a university is simple: it is a place in real life which is equal to a monument around people are grouped. Such a monument makes in the internet world no sense, because the internet is not dependent on physical locations. Physical instances of information are there, for example the Google datacenter or the AT&T cable, but these places are not called universites, they are called datacenter and fiber-optic line.
The result is a misunderstanding. If someone is asking for money to build a library, the other side will not understand him. But if somebody is asking for money to build a fiber-optic connection he will be understand very well. What the Open Access debate is about are words. The idea behind Open Access is to make clear to the public what a library is. The danger is high, it will become forgotten.
Museum
Sometimes, the youth is accused, that they are not interested in books and schools. That is totally wrong. Especially the youth is very interested in printed books and teaching environments in which they must be quite for 45 minutes. The difference is, that they are visiting museums for getting more information about it. Digital natives who are familiar with tablets and internet access are highly motivated to visit a place, in which they can see the Koenig&Bauer press from 1811, or to attend a play in which a classroom has a classical whiteboard and a physical teacher. Taking a look backward is very helpful to understand the world. The only thing what the digital natives are not understand is if somebody is not aware that the past is over, that the things have changed over the years. Watching a gutenberg printing press in a museum might be funny, but using it for printing out a real book is a waste of time.