Page templates are the most open-ended template type. They can serve as any flavor of web page or landing page.
Partial templates are coded files that can be included in other templates. This allows developers to take a more modular approach to building complex documents and sharing markup between multiple files.
Both page and partial templates can contain Liquid markup and perform complex processing. They can also have access to all global, relative, and variable objects, whether defined explicitly or available implicitly based on the page and object being rendered. Partial templates get compiled into templates when published.