'Suitcase' is an object - it's large (around 70 cm height), flat, rectangular, always with handles and sometimes with wheels. Grammatically, it is a COUNTABLE noun. You can say 'a suitcase'. You can say 'How many suitcases do you have?' 'I have two suitcases.'
Baggage and luggage have the same meaning. They are general terms to describe everything you pack your belongings in when you travel. Suitcases, backpacks and bags are all examples of 'items of luggage'.
Grammatically, these words are UNCOUNTABLE nouns, which means that we say 'How much baggage/luggage can I take? 20kg'. We can't say 'a luggage' or 'a baggage' or 'luggages' or 'baggages'.
The only difference between these words is that 'luggage' can refer to suitcases and bags when they are empty. You can buy new luggage before you go on holiday, for example.
NB Another difference is that you can use the word 'baggage' figuratively, as in the term 'emotional baggage'.
And that's it. That's the difference.