The Node Browser is primarily an administrative / developer tool that shows the "raw" structure and contents of the repository (hence the reason it's only accessible by administrators). In contrast, the Web Client displays a user friendly view of the same underlying data structures.
The search feature in the Node Browser allows one to execute native searches against the repository using the various query mechanisms supported by Alfresco (XPath / JCR, Lucene, etc.).
Cheers,
Peter