A search engine uses various programs, webbots, to catalogue web pages on the internet. Not just sites but individual web pages. This compares to the index of a book. Directories, however, are more like the table of contents. A web site rather than each page is submitted to a specific position in the directory.
Directories are great for browsing similar sites.
Search engines are generally more comprehensive and allows you to find content that may appear on only one page of a web site. The following lists the major search services and their primary structure.