In terms of a site search, if you have a database-driven site, then any of the searchable information in the DB should already be embedded in the page. The only reason most sites' internal search engines search the database is because it's more convenient, as spidering, parsing webpages, interpreting their semantic structure (title, headings, content, etc.), and then indexing the pages is a complex and difficult process. OTOH, a database is already indexed and its structure provides semantic information about the data.
However, Google has already solved the problem of spidering, parsing, indexing and searching webpages. So if you have properly marked up, semantically structured webpages, then it should make no difference. Google will probably offer even better/faster results than what you yourself can implement using full-text search just because of their synonym replacement & natural language processing algorithms, advanced search operators, and other features.
Now, if you're trying to implement something like a VIN search, or a highly specialized search engine for a particular domain (e.g. an industry knowledgebase), then you may be better off implementing your own database search engine. But then you're not implementing a site search anymore.