I spend a lot of my time trying to explain the difference between both but I find that sometimes it is easier to use an example. So lets try this video… from the amazing OK GO
UI is the song, but UX is the video. Would you have the same experience if you heard it on the radio?
Or another way, UI is a candle, but UX is the romantic dinner in the photos below
The great folks over at http://designmodo.com/ux-ui/ make the comparison that UI is a simple spoon, perhaps even a beautiful spoon. But UX is satisfying a hunger, maybe the spoon make the meal more enjoyable but it is only a small part.
Finally here is a really good list from http://www.uxisnotui.com/ that has grown legs in UX communities that lists the key differences between UI and UX
What do you think?