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Outside of the emotional realm (as noted by FumbleFingers), they both generally refer to "that stuff you're traveling with". I think I'm more likely to say "baggage" for irregular items (e.g. golf clubs) and "luggage" for things packed in suitcases. "Luggage" but not "baggage" can also refer to the suitcase itself; when you buy a suitcase you're buying luggage, not baggage, but when you fill it up and get on a plane you're carrying either luggage or baggage.

The basic difference is that "baggage" has a greater scope of meaning than "luggage".

Dictionary.com on "baggage":

1. trunks, suitcases, etc., used in traveling; luggage.


2. the portable equipment of an army.


3. things that encumber one's freedom, progress, development, or adaptability; impediment

Dictionary.com on "luggage":

1. suitcases, trunks, etc.; baggage.

It's interesting that there are a couple of answers here that suggest "baggage" is more general than "luggage" - I would put their meanings the other way round, with luggage being more general than baggage.

In broad terms, based on the roots of the words,

Baggage = stuff in bags

and

Luggage = stuff you lug (= carry/drag)

So irregular items that don't fit into bags are luggage but, strictly speaking, not baggage. You might even go as far as to say that your luggage consists of your baggage plus any unbagged items you're taking with you. Nowadays however such distinctions are somewhat pedantic/archaic and the words are more or less synonymous.

Well, the place where you pick up your luggage at the airport is normally called Baggage Reclaim, so they are pretty much synonyms.

Except if you're talking to your therapist, for example, who may say you're carrying a lot of 'emotional baggage' (sense 3 here). But I never heard of 'emotional luggage' - maybe I've been seeing the wrong kind of shrink.

Possible your sense of baggage being more 'general transportation' is because you have some awareness that in a military context it covers just about everything the men can carry without needing transporters.

To "Lug" is to drag/carry a heavy object. Thus big items like suitcases and big bags on wheels and all other big heavy stuff were called luggage. Smaller bags and stuff you could bring with you on your person all the time was called baggage.
Nowadays those distinctions are gone, and the two words are use interchangeably.

I actually remember reading that the difference between the two is down to the US/UK divide.

In British English, "luggage" is preferred usually. While in American English, "baggage" is preferred usually. However the two words are being used more commonly on both sides of the pond, especially with the word "baggage" at British airports due to US travelers.

Baggage is more understandable than luggage to a foreigner.

I would say from working with Brits and Americans that luggage is more often used by Americans. I rarely if ever heard them using the word baggage. So I'd rather go with the fact that they are synonyms but luggage is rather American English whereas baggage is British English usage.

"Baggage" or "Luggage"?
There seems to be some confusion as to which word is more ample: luggage or baggage.
In my mind it's very clear that "baggage" is the more extensive concept.
For example, armies don't carry luggage. All their equipment (where clothing is not even the most important item) that travels with them is always baggage, never luggage.
And who has ever heard of a troubled person carrying "emotional luggage"?