The only difference I can think of is that UI may include looking beautiful (gradients, shine), while UX isn't concerned with that. Really, you're talking about similar things.
A user interface is different from the user experience because of the interaction (abbreviated Ix; interaction design is IxD).
User experience is the entire experience of the user, and you can broaden it to include the documentation, the packaging, the APIs, the phone menus (I do), or you can narrow it to include just the purview of the application itself.
The user interface is just that: the interface, the actual buttons you click on and command prompts you type text into. The interface is controls laid out by developers and front-end engineers plus a visual treatment defined by graphic designers and visual artists, but the user experience designers told them what visual and contextual priority some controls have over others, provided a recommended information architecture for menus, a expertly defined placement for widgets, suggested reaction times for responses and copy for dialogs.
The study of how the users will interact with the application is the difference between throwing a bunch of controls in a window, and designing the experience they will have.