Basically the difference is to do with security.

With Private folders you need to log into the website to view.

With Hidden folders you dont need to log into the website to view but you will need to know the URL.

For example

If you apply a folder to Level 2 of a menu and you dont want it to be viewed in the menu.

The Folder option you will choose is Hidden.

If you apply a folder to Level 2 of a menu and you want the user to log into the site view.

The Folder option you will choose is Private.