data·base: noun 1. A collection of data arranged for ease and speed of search and retrieval.
--The American HeritageŽ Dictionary of the English Language: Fourth Edition. 2000.
This quality of being " ... arranged for ease and speed of search and retrieval" is what distinguishes a database from a computer network like the Internet, which has no standardized organization principle.
Databases may sometimes be accessed through the Internet, but their contents are not retrieved by search engine services like Google or Yahoo! Most are available through separate Web sites that charge a fee for use, normally paid by libraries on behalf of their users.