What's the difference between private message & message?
By using the Message module, you can create your own custom "Events logging" (to register things that happened in a site).

A great sample of this you can find in Commerce Kickstart, which uses the Commerce Message module, to create such "Events logging". In this case in the format of an Order History, as shown in this screenprint (from this module's project page).

The Message module is an amazing module, though there is not a lot of documentation about it. It is fully entity based, which implies that it perfectly (out of the box) integrates with modules such as Views and Rules. And to give you an idea of the kind of things you can use it for, have a look at the answers to:


How to create a basic message - notifications system displaying a real time message counter badge?
How to show message view containing own and following activity?
How to integrate messaging into a custom module?
Is there a simple way to notify users about new posts on the site?