When paired with ない (either the い-Adjective
or the auxiliary verb), the 副詞（ふくし） (adverb) 別（べつ）に ‘separately’ changes slightly in definition to that of 特（とく）に ‘especially’. Let’s have a look at a few examples.
I actually don't mind, but my dad cares a lot about things like that, so please don't say such things in front of him.
It's not really painful. It just stings a little.
It's not that I don't like you at all. It's just that I don't know what to talk about with you.
Everyone is calling it a party, but it is actually not that big of a deal.
This ‘especially’ nuance of 別（べつ）に comes from ‘separately’ being negated. The literal meaning of 別（べつ）に～ない is something like ‘(A) cannot be separated/differentiated’. This comes across as ‘maybe it is a little (A), maybe a lot (A), but in any case, it cannot be said that it IS (A)’.
別（べつ）に will quite often be paired with わけではない
, as they both express uncertainty. However, it should never be paired with わけがない
, as this would be combining a statement that removes certainty, with one that adds certainty.
It is not that I dislike cakes. I just don't want one right now.
It is not that I SO don't not like cakes. I just don't want any right now. (Grammatically correct, but very confusing for the listener)