The free Payage Component does all the hard work of generating payment buttons and managing payments. But Payage needs a "client", or a "Payage application" to tell it how much to collect, and for what.
The Payage Plugin is a Payage application that makes payment buttons in a Joomla article. It uses Payage to collect payments. It is not part of Payage and is not required for Payage to work with any other Payage application, like Flexicontact Plus, MediaShop, or Rentalot Plus.
The Payage Plugin is completely optional. It is small and light and quick to learn, but has limited capabilities.