All the words mean ‘brown sugar’ but there’s different nuances and contexts.
흑당
This word is only really used to refer to the brown sugar milk tea flavor.
설탕
I’m dropping in a bonus word. 설탕 just means ‘sugar’, but is used to refer to what we call ‘white sugar’.
흑설탕
This is the most common word to refer to ‘brown sugar’. It seems to be the basic brown sugar that we use in cooking.
I have had one Korean tell me she uses 황설탕 for cooking and 흑설탕 for face packs, but everyone else seemed to use 흑설탕 for cooking.
황설탕
This is also brown sugar, but a darker kind than 흑설탕 and seems to be used less often.
Satyanghae Korean 