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As I understand, they are both very close in terms of meaning.

E.g.

> 私は歌を歌えない **わけではない** が、人前で歌うのは恥ずかしい。  
>  私は歌を歌えない **ことはない** が、人前で歌うのは恥ずかしい。

Both sentences would mean: "It's not that I cannot sing, it's just that I get
embarrassed in front of others."

I'm pretty sure there is a slight difference in nuance, that I can't quite
grasp yet.

Is it a matter of opposing the belief of someone that you can't? (while the
other is plainly a statement?)
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question
grammar word-choice nuances
What is the difference between (ない)わけではない and (ない)ことはない?
There is very little difference, and they are interchangeable most of the
time. Still, I feel the speaker is more aware of the previous context when
わけではない is used. That is, a sentence using わけではない tends to have a nuance of
"that is not to say", "that doesn't mean" or "I'm not saying". ことはない is a
relatively "plain" double-negative sentence.
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answer
Not sure if this is what you are looking for, but -タ form seems to behave
differently for the two constructions. As I see it, なかったことはない coincides with
another construction -たことがある, which takes precedence. なかったわけではない doesn't have
the same problem.

> 私は歌が歌えなかったことはないが、人前で歌うのは恥ずかしかった。

Most people will read this as "I have never been unable to sing" (="I always
could"). The other interpretation ("It's not that I couldn't sing"="I could to
some extent") might not be impossible, but it would be much harder to read
that way.

Whereas, the same doesn't happen when you change ないわけではない to なかったわけではない.

> 私は歌が歌えなかったわけではないが、人前で歌うのは恥ずかしかった。

The only difference from the original sentence is that it describes a past
event.
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answer