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.
Christian Poser the 28 year old four man team gold medalist at 2010 Calgary world cup has been in the bobsled competition since 2008. Christian was born in Cottbus Germany on 16.8.1986. He has represented the German team in several championships namely the world cup of 2013 at St Moritz, the European championships at Igls, the world cup 2012 at Lake placid and the world cup 2011 at Koenigsee. Standing at a height of 6 feet and weighing 95 Kg Poser is a respected member of the German squad. He is also engaged to the US bobsledder Jamie Greubels.