What Is The Difference Between 'Neither Of Them' And 'None Of Them'?

Nothing, I don't think that there is any difference.

I can see that this is amazingly complex, but I might try to do it for you.  Neither means no one or the other.  So one might so something along the lines of:

I asked is she wanted coke or diet coke, and she answered neither.

Here neither means that neither are wanted from the two choices.  None of them suggest that it is not any of the choices that are wanted, no specific any choices but specifically not one of them.

To use this in a sentence, one might be say something along the lines of:

I asked if Claire liked any of the designs, she said that she liked none of them.  (this could be hundreds).

None can could be say, three to a million, billion, trillion, but neither would represent two and only two choices.

I hope this help, I think I wrote it a bit confusingly!