Luggage refers to the physical thing that you pack when travelling. Baggage is pretty much the same, except it can be used in a broader sense and also metaphorically. Often people can be referred to as "baggage", because they're 'heavy' or 'limiting' in a way. You might also hear "emotional baggage".

In the U.S. they are pretty much interchangeable synonyms, with exactly the same usage rules. As Andrew says, when used figuratively to mean difficult burden, we use "baggage." I can't come up with any explanation of how we choose one or the other. Googling suggests that airline use the word "baggage" much more often than "luggage." "Luggage" is a little dignified--you can say "expensive leather luggage," but "expense leather baggage" doesn't sound right. "Baggage" emphasizes what it is--a container, like a bag--and "luggage" suggests what you do with it--lug it.

Andrew has beaten me to it,and written a very good explanation!