Young woman works if you're old enough to call an adult woman "young." If you're 20, saying young woman is just awkward. "Young lady" really conjures up images of a child. "You watch your mouth, young lady, or you won't get any dessert!"
I agree with Susan B that women are infantalized, but I think "girl" has evolved to not have that connotation anymore. "Little girl," on the other hand certainly still does. http://www.youtube.com/watch?v=u0SSeACInqw
It's like "you" and "thou." "Thou" was originally the familiar second-person pronoun, and "you" was formal. The words have evolved so that now "you" is quite informal, and "thou" is considered so formal that it can't be used without sounding pretentious (thanks, King James!).
I think the whole use of girl vs. woman depends on your audience, and in most contexts it is obvious you're not referring to child. In an informal setting I will tend to use girl to refer to any female not significantly older than me, and will use guy for any man who isn't a young kid. In a formal setting like in front of a judge, of course I'm going to use woman and man or gentleman.