You aren't even comparing apples and oranges here. This is more like asking what is the difference between apples and apple pie.
A Page Object Model is a way of representing an application in a test framework. For every 'page' in the application you create a Page Object to reference the 'page'. A Page Factory is one way of implementing a Page Object Model. Search for "Java Factory" to understand the concept of a Factory.
Additionally, the Selenium project has a
PageFactory implementation. This is an actual implementation of a Factory rather than just the concept.
Darrell