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.

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Manahel Thabet

In 2008, aged 25, Yemeni economist and scientist Manahel Thabet became the youngest person to receive a financial engineering Ph.D. magna cum laude. Thabet earned the degree at the University of Illinois and has since worked towards a second Ph.D. in quantum mathematics. In 2012 she came up with a revolutionary 350 page formula to calculate distance in space without the use of light. She also has an IQ higher than 168 and garnered a Genius of the Year Award and a spot on the 2013 World Genius Directory s list. Thabet set up the company Smart Tips Consultants in 2008. Her accolades include a humanitarian award for her efforts with the United Nations and a Woman of the Year Award from the Women s Federation for World Peace.