What are the differences that can be found between doing research at the library and doing research on the internet?
To some degree, differences between library research and on-line research are dependent upon the quality of the library being used as a comparison. But generally, for a good university library and on-line research, there are some differences.
First, even with access to a good database, there are likely to be many instances in which you do not have access to the full article, just being able to read the abstract for the article. This limits your research considerably, and it means if you find an article that you really need, you will need to go to the library to read it. Most good university libraries subscribe to the actual periodicals, which means if you go to the library to begin with, you will have access to more.
Second, because of the nature of the search engines we use, our ability to "browse" is severely limited on-line. If I go to the library and look at periodicals in my area of interest, I can browse through the hard copy and often see something of interest to me. The serendipity of library research is one of its great benefits.
The best research, in my opinion, is both kinds of research, getting the benefit of all that is on-line and the benefit of the more complete collections in a library, along with the browsing experience.