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.
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私（わたし）は別（べつ）に構（かま）わないけど、お父（とう）さんはそういうの気（き）にするから、お父（とう）さんの前（まえ）ではそういう事（こと）言（い）わないでね。
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.
別（べつ）に痛（いた）くないよ。少（すく）しチクってするだけ。
It's not really painful. It just stings a little.
あなたの事（こと）は別（べつ）に嫌（きら）いではないよ。ただあなたと何（なに）について話（はな）せばいいか分（わ）からないだけ。
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.
パーティーってみんな言（い）っているけど、別（べつ）にそんなに大（たい）したものではないよ。
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.
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別（べつ）にケーキが嫌（きら）いというわけではない。今（いま）は食（た）べたくないだけ。
It is not that I dislike cakes. I just don't want one right now.
別（べつ）にケーキが嫌（きら）いというわけがない。今（いま）は食（た）べたくないだけ。
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)