These words are the same. Both of them refer to the collection of suitcases/bags you take with you while traveling.
Both of them are uncountable nouns, so don't use "a" or make them plural:
I have three luggages.
I have three pieces of luggage.
I accidentally left a baggage at the hotel.
I accidentally left a suitcase/bag at the hotel.
I accidentally left a piece of luggage/baggage at the hotel.
The place in the airport where you pick up your bags after arriving at your destination is called the baggage claim. But when the airline loses your bag, we usually say they lost your luggage (not baggage).
The word baggage has an additional meaning: emotional or psychological effects from the past that negatively affect your present situation. For example, if you had a past romantic relationship with someone who lied frequently and now you have problems trusting anybody, you have "baggage" from that relationship.
This triple juice drink is a refreshing nonalcoholic option for entertaining. Just mix up a pitcher just before the guests arrive. The three juices?orange, pineapple, and cranberry all contribute disease fighting vitamin C.