User experience (UX) is, quite simply, the feeling that one gets from using a system. Did you accomplish what you set out to do? Were the features of the system intuitively obvious? Was the system designed to improve your day and make you more awesome?
UX is the intangible design of a strategy that brings us to a solution.
User interface (UI) is how you interact with a system. Are you pushing buttons, pulling levers, or typing on a keyboard? Are you finger swiping a screen, giving voice commands, or using an eye tracking device?
Certainly, a badly designed UI can leave a consumer with a terrible experience. I can think of two Top Gear episodes where Jeremy offers money if a person can get into a car and start it. If the door handle is in some obscure spot, the consumer is frustrated. If the ignition is difficult to locate, the consumer feels stupid. UI design patterns exist so that finding basic functionality isn’t a guessing game.
Trader Joe’s has a revolving display of vintage drawings on their credit card terminals. The artwork is not necessary to the check-out experience. But, it is a lovely UI that kept me entertained and left me with a “well, that was fun” feeling.
A really good UX can entail user research, prototyping, information architecture, taxonomy creation, and usability studies. UX provides the solution. UI is what the customer sees on the way to reaching that solution.