Here are some of the advantages of WPF over windows forms:
1. A designer can easily work on the UI while the developer focuses on the application logic.
2. The content may flow, like contents of a web page, depending on the layout and properties of containers.
3. Support for XPS printing. XML (XML Paper Specification) basically prints better than using GDI.
4. In WPF, you draw things using shapes and other graphical objects, not pixels.
5. There is support for commands, wherein you only need one command that can be called in different places like menu items or context menu.
6. There are data binding and dependency properties that greatly helps in creating a loosely-coupled design between the presentation and business logic layers.
7. Routed events allow you to centralize event handling code or prevent an event to be propagated along the hierarchy of controls.
8. Because of resolution independence, you don't have to worry about the behavior of the application when the resolution changes.
9. Data binding to LINQ queries is supported.
10. You can also use windows forms inside WPF controls.
I believe there are many more advantages, although I think that its main advantage really is the user interface.