Due to Yahoo! and Google previously offering a search and directory function, sometimes the two services are intermingled and confused for one another. Ultimately, however, they bear no resemblance, apart from the fact they both attempt to provide users with relevant hyperlinks for any given subject.
The web directory is a hierarchy structure of Web pages; each directory page will contain a list of hyperlinks on a specific subject. Dmoz.org is the best example of a functioning directory. During the Web's infancy in the early 1990's search engines had difficulty in differentiating between relevant content and spam. Directories, on the other hand, tended to use humans to pick and list the hyperlinks on their pages. This helped improve relevancy, but can lead to dead links, abuse and link rot over time, which is hard to keep on top of for very large directories.
The search engine is a web page with a search box; a user will input a query into the search box, and a script will search a database to find hyperlinks which are the most relevant to the query. Due to the improved results search engines are providing, they have become the most popular way to find content on the Web. Search engines use a robot/crawler to fill it's database; a crawler will start at a designated web page, and will then follow any hyperlink found upon that page. Search engines use an algorism to rank web pages for their quality of content; the algorism is the key component for a search engine to provide relevant results and to weed out spam.
In conclusion, while directories were a very popular way of finding content on the World Wide Web: as time has passed, search engines - such as Google - have easily surpassed them in terms of usage and overall user satisfaction for finding relevant information.