Both are correct: "pół jabłka", "połowa jabłka". It's the difference between "half an apple" and "a half of an apple", I'd say.
You have something different here: "połowa jabłek" - "half of the apples" (half of the apples they had in the store, probably). "Pół jabłek" would be wrong.
Is połowa a different case or a different word with the same... root word?
Different words, even different parts of speech. "połowa" is just a noun, but "pół" is a numeral.
I think the English translation sounds a bit colloquial here by clipping the sentence and omitting the word 'of'. I would probably say 'I am buying half OF the apples' here. I know we readily omit small words and connectives in everyday speech but I probably wouldn't in this case.
I am relieved! I thought the sentence was: Kupuję pół jablka. And that seemed pretty twisted. Poor dude!