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".