From Wikipedia: User Interface (http://en.wikipedia.org/wiki/User_interface) - The aggregate of means by which users interact with the system (e.g., all the means of input and output)
User Experience (http://en.wikipedia.org/wiki/User_experience) - The architecture and interaction models that impact a user's perception of a device or system ("all aspects of the user’s interaction with the product").
UI is the "how" of creating an interface (implementing shiny controls, that sort of thing) and UX is the "why" (creating a good experience for the user).
What's interesting is that there are many counter-examples... for example Craigslist. Terrible UI/UX, yet it is wildly more popular than it's closest competitors. For a while Plentyoffish was the most popular dating site, same thing there.