Unlike usability (which is defined in a ISO norm), “user experience” is used somewhat loosely across the industry so that it's difficult to claim that any particular definition is the definition of UX.
What can be said is that the notion of user experience appeared much later than usability, in an effort to go beyond purely utilitarian aspects. In that sense, UX therefore includes aesthetic qualities and the fun or pleasure one can derive from using an interface or product and is broader than usability.
It's quite common to see UX as a step beyond usability, a way to differentiate a product from its competition by providing some intangible hedonic value above and beyond fulfilling a given function efficiently but it's possible to imagine products that elicit a superior user experience with only average usability. I don't have a very good example but one that come to mind would be early real-time strategy games like Dune 2000. They were extremely well done and many players loved them but you had to move dozens of units one by one which is a real pain if you compare that to the usability of later games.
It's less common to present things in the other direction but in some ways, usability is also broader than UX. If you look at the common definition, satisfaction, the “subjective” part of usability, is only one of three aspects of usability (together with two “objective” aspects, effectiveness and efficiency). It's therefore conceivable to experience a product as usable and very easy to use even if it's not the most efficient one and some other interface can be shown to enable users to complete a given task quicker.
In practice, user experience, satisfaction and other aspects of usability are often related but they are all slightly different ideas. If you want an academic take on this, you might want to look at http://www.allaboutux.org/uxwhitepaper
There is in fact a number of academic papers looking at the distinction between perceived usability and aesthetic quality (which can be thought as a part of user experience) and debating how independent they are empirically. See for example:
- Hassenzahl, M. (2004). The interplay of beauty, goodness, and usability in interactive products. Human-Computer Interaction, 19 (4), 319-349.
- Tractinsky, N., Katz, A.S., & Ikar, D. (2000). What is beautiful is usable. Interacting With Computers, 13, 127-145.