For me using both words quite long, in both languages Croatian and Serbian there is no difference , both express the same feeling. Maybe in Croatian living with common people I have heard more "tuga". I was searching the origin of the word "tuga" and found it in a legend about 2 sisters Tuga and Buga and 5 brothers who were leading Croatian tribes to Balkan, Dalmatia. One legend says that Tuga was nice and happy and people were also living happy life under their rule (they were Queens) After their death there was a sorrow going from generations to generations.
But another legend from Bosnia is saying terrible things about these 2 sisters who were spreading great disaster and sorrow on their way through Bosnia. I was also shocked now. ( legend about Tuga and Buga ,BOBOVAC-LIST VARESKIH HRVATA (BR 36. PROSINAC ´97)) Probably the last legend can be the source of this word.
Otherwise as I know these both words are synonyms at least in Croatian, Serbian, Bosnian, meaning the same feeling-sad
Both words are synonyms as Onitamo said but the word "tuga" is more about feeling while the word "žalost" is more trivial. For example, the day of grief is "dan žalosti" never "dan tuge" or when someone dies, the closest family members are in mourning ("žalosti") for a year and during that period wear black clothes, but grief ("tuga") for the dead can last forever...When someone wants to apologize he can say "žao mi je" but not "tužno mi je".