In general, flat-leaf parsley has a more robust flavor, while the curly variety is associated with decoration. Some claim that curly-leaf parsley has no flavor or, conversely, that it tastes more bitter, but it really depends on the particular plant, its growing conditions, and age. Both kinds of parsley may be used in cooking and when substituting one for the other, taste to determine the flavor and adjust as desired. Think, too, about the texture that would work best in your dish.
Finally, don't discard the stems, which have a stronger flavor than the leaves and can be used in a bouquet garni and added to homemade stock or a pot of beans.
Related: Humble Herbs: In Praise of Parsley