In the File Entity module, what is the difference between "Manage display" and "Manage file display"?
Manage display is the core page to manage the display of fields for various view modes.

The Manage file display tab is provided by the File Entity module; the page callback for that tab is file_entity_file_display_form(), whose description is:


  Form callback; presents file display settings for a given view mode.


So the difference is that display settings that relate to the file entity module have been separated out from the core Manage Display form, on a per view-mode basis, and that extra tab is where you set them.

As noted in the comments this is covered in a lot more detail as part of a broader question