Although we've been making some distinctions between the Web and the library, the two aren't entirely different things. It's important to understand that there is a middle-ground -- the idea of the "library on the Web." That is to say, many libraries have web sites which organize information and provide access to collections of quality resources.
One great thing about using the library on the Web is that the information has been evaluated and organized. Much of this information is from the government, companies, universities and foreign countries. Sometimes the library has digitized part of their own collections for people around the world to use. Keep in mind that although there is an increasing amount of information in this "digital library" you will still not find electronic full text versions of all the resources you would find in the physical library.
Another aspect of this library is how easy it is for you to access. Library web sites often have information about library hours, policies and contact information if you need assistance. If you are a student at a university, you can use the library online 24 hours a day, seven days a week from any Internet-connected computer. In some cases you can even find the full text of articles from magazines and journals for the past 20 years all through the library's Web page.
One of the best resources available from the library over the Web is the periodical index.