Saying "there are not so many apples" kind of sounds like there was an expected amount to begin with and you're coming to the realization that there aren't that many.
Saying "there are not many apples" is just a general statement of the fact that there aren't many in front of you.
Just my opinion.
There aren't really any apples.
That's how it should be translated with 별로 (really) acting as a downtoner of "not"; i.e. in the same way as "Not exactly " is used to soften the meaning of "No".
DLG answer: "There are not many apples" = 사과는 많지 않습니다 ? ('Not many' seems to imply there are some, though not a lot.)