The Internet is truly not a library, and shouldn't be confused as one. Ask most students what a library is, and they inevitably offer a reply with the words books, information, borrow, etc. included. If you truly understand a library, which in its essence must have at the very minimum, these words- information, evaluated, selected, categorized and organized for access- then the clarification becomes obvious.
In Siva Vaidhyanathan's 2010 publication The Googlization of Everything (And Why We Should Worry), he offers a very clear explanation of the mighty search engine (and everything else) we call "Google". The Internet and the dominant search engine of the day, is different in all respects from a library.
We need to ensure our students are aware of the many differences between the Internet and libraries. We don't need to "worry": our students need to be informed, and aware, of these differences, and develop competencies in searching, evaluating and selecting, recording, processing and synthesizing the information into knowledge from both online and print sources. And let's not forget, most libraries include access to the Internet, and online databases. It's not an either/or choice. Rather, it's an informed and integrated selection of the best resources for each specific information need, and best suited to who is doing the research- age, learning style, etc. all come into play. It definitely isn't a simple choice of library OR Internet. Our world is a complex, burgeoning world of information, non-information and misinformation. We, and our students, don't need to "worry" about that, we need to be informed, and astute consumers of information, from all sources.