A saved search stores the SPL query so you can re-run it without retyping. You can share saved searches with other users by setting their permissions to "App" or "Global".
Reports are saved searches that have been given a title, description, and optional schedule. They live under the Search & Reporting app's Reports tab.
When saving, you choose the permissions: Private (only you), App (users of the current app), or Global (all apps).