Traditionally UI is just the visual presentation layer and UX is the action of interacting with something. For example, when sorting through large amounts of data (like am amazon search results) a site might let you narrow down your search results using an additional set of criteria (like price, cost, etc…) This is is a UX pattern that is sometimes called a faceted search (http://patternry.com/p=refining-search/).
Note, however, this no where is it mentioned that this needs to be a vertical list with checkboxes or sliders. Faceted search is often represented this way however, because that is a common UI pattern (http://tabpatterns.com/pattern/filtered-search/).
Hope that helps you understand the difference.