I'm not an expert on NoSQL, but based on general computer knowledge I would say some advantages over a filesystem would be:

  1. Single entry point.
  2. Not having to share your filesystem.
  3. Centralized security separated from your OS
  4. Simple key/value access (no folder structure)
  5. Abstraction of the filesystem (like #2)
  6. Distribution of data over multiple nodes.
  7. Separation of NoSQL admin and network admins.
  8. Shiney new thing :)

I'm sure now a days, an advanced filesystem/sharing protocol would handle some of these.

For one, should be faster for small operations.

Some databases also let you define secondary indexes.