Where should you search? The web or the library?
Both!
There are things freely available on the web that you won't find in the library. Many governmental, nongovernmental organizations (NGOs) and International Government Organizations (IGOs) post reports and sometimes data directly to the web. There are some amazing academic projects and important blogs.
There are also sources (journals and news sources) that post a few articles to the web, but then require payment for additional content. Google Books, of course, may have a preview (or sometimes the full book) of an ebook that the library doesn't have.
On the other hand, there's an enormous amount of material, not freely available on the web, that is only available through the print and online resources the library purchases -- books, ebooks, e-journals, data sources, news sources, government documents, primary source materials, streaming media, and more.
The library catalog and databases also offer ways to refine and filter your search that are not available in any web search engine.
Experienced researchers use both tools, and go back and forth between the two.
Differences between searching the web and searching the library
Search engines don't want us to think too hard about how we are searching. We type in our question and sometimes we find information that seems to match the question.
That's nice, but it doesn't always work as well as we hope. Unfortunately, the algorithms used by search engines are proprietary (trade secrets). We don't really know exactly how they are processing our search terms -- when they work and when they don't.
The Library Catalog, Advanced Search, and the library databases, process words and phrases connected by "operators." The operators are the terms AND, OR, NOT. The terms are matched to titles, to summaries of the books, tables-of-contents, and sometimes to the full-text of the book. That's how it works. No secrets.
In addition, while you may find journal articles, via Google Scholar, (but use Passkey to connect to what the library has!), it is less common to find full-text ebooks or chapters on the web.