Translation:I am washing cups and you are washing plates.
"Чашка" is a cup, "горнятко" is more like a mug, a bigger version of a cup, usually. But it is used interchangeably as well. A beer mug will be "кухоль" though. Wine glass is "келих". For my Western-Ukrainian ear "бокал" sounds more like Russian contamination, but it is also used frequently.
I believe the main difference between «ча́шка» and «горня́тко» is the language history. «Ча́шка» is the older word, «горня́тко» is the newer one.
Ча́шка has been the most-used word for 'cup' in the past.
The word горня́тко used to be dialectal, and has only recently became acceptable in literary language. The 1970—1980 dictionary still says that «горня́тко» is same as «горня», i.e. 'small pot'.
My usage experience tells a bit different story. The words are used interchangeably, but with "горнятко" you still imply a bigger size, than with "чашка". We use "горня" a lot as well, in the same sense as "горнятко" (but a still bigger size may be implied with it too). I've never heard anyone using it in a "small pot" sense though.
But note that I'm native to Lviv, and word usage here in Galicia may differ considerably from the rest of the country (things get even weirder if you go to the Carpathians, and weirder still if you cross them to the Transcarpathia). "High", or what we usually call "literary" Ukrainian (as it is in the classic Ukrainian literature) was based on Central-Ukrainian dialects. Would be interesting to hear about usage experiences of people from other parts of the country.
Note also that Soviet dictionaries are not a totally unbiased source considering the word usage in real life. You may google about the language reform of the 1930s if you want to know why.
I'm also originally from Lviv, although I've lived most part of my life in Belarus. I don't feel much difference between горня́тко and ча́шка. My Ukrainian relatives used горня́тко very small cups. I still keep the cups my Grandma bought — they are smaller than regular cups, and I'm pretty sure she called them горня́тка.
Note also that Soviet dictionaries are not a totally unbiased source
Older dictionaries agree with the dictionary I've linked. Here's what Hrinchenko's 1907–1909 dictionary says:
Горня́тко, -тка, с. 1) Ум. отъ горня. 2) Въ хороводной игрѣ вербовая дощечка трое играющихъ малыхъ дѣтей. Kolb. I. 159. 3) — пщі́льне. Ячейка у пчелъ. Вх. Лем. 405.
Горня́, -ня́ти, с. Горшочекъ. У печі палає полум'є на припічку, на жару горшки й горнята. Левиц. І. 37. Ум. Горня́тко. Єсть молоко, — буде і горнятко. Ком. Пр. № 429.
You may google about the language reform of the 1930s if you want to know why.
I know about the language policy in Soviet Union, and I don't need to google this. Your sentence assumes my ignorance, and I find this condescending and disrespectful. Please reconsider your tone.
I wrote it not for you specifically, but to give the information in general. And no, not all of the Ukrainians know about it as well, and there are still really a lot of people who see USSR in quite a positive light too. As I don't know your background, how can I possibly know that you know about this moment in Ukrainian history? I mean specifically the language reform and the language policy. I work as a tour guide and meet a lot of people from all over the country, and most of them have a very vague idea about Ukrainian history (usually it is a consequence of bad education which is still a common problem here). So I just wrote there was such a reform in case you didn't know, as it's too big a subject to elaborate on it here. No offence was meant.
Thank you for the dictionary citations. I guess, the "little pot" usage is obsolete now, at least in Lviv region, haven't heard it from anyone.