The difference is simple, yet it makes all the difference in flavor. Where as normal paprika is crushed dried chilies, smoked paprika uses chilies that are are smoke-dried and then crushed. Most often the peppers are Pimenton de la Vera and are typically smoked with oak which leads to a strong outdoorsy flavor.
We have found that once you become familiar with the smoky taste it is an easy step to using smoked paprika in interesting ways. Try sprinkling some on top of casseroles (especially potato dishes), add to meat marinades or go crazy and put a teaspoon or more in your favorite peanut butter cookie recipe! Delish.