User experience is something we understand, support and design. User interfaces are small windows into the total experience. Here's a story/metaphor I think is useful to explain the difference:
You are walking past a storefront, and you see an item in the store window that catches your interest. You stop, and look. Next to the item is a sign that details additional information about the item and the price. You enter the store, and look around at the surroundings… It's nice, clean and you are immediately greeted by a friendly person who asks if they can help you.
"Not now, thanks." you reply, almost automatically… And the person who works there smiles, and replies that "If you have any questions, my name is Joe, come find me… I'm here to help."
You walk on, and unconsciously note that the person you just interacted with was kind, friendly and was there to help… Not just to sell you something. You see a couple of displays that show some interesting products. One of them allows you to play with the item. You pick it up, turn it on and use it. After a few moments, you find the item in the window that drew you into the store.
As with the product just before, you can play with it. You lift it, interact with it, and look at the packaging to see more details of the item. You decide to buy it.
Item in hand, you walk towards the checkout. The person behind the counter says, kindly, "Did you find all you were looking for?"
"Yes," and you complete the purchase by punching your PIN number into a machine after you scan your debit card into it.
"There you go." Purchase completed you leave the store. You leave to continue your day.
There are only four "user interfaces" in this story – the window display, the two product displays, and the screen the user engaged with to enter his PIN number. Only one – the ATM device at the checkout – is an actual "screen." The whole story details the "user experience" that the customer has…
And even then, the user experience is more than that. Why did the customer buy the product at all? What drives them and what are their mental models? That is the focus of user experience design, to create an experience that supports the customers needs at all points of their engagement with your products or systems.
User interfaces are part of that engagement, but sometimes is a very small part.