Qual è la differenza tra She won't change her mind in any case. e She is not going to change her mind in any case. ?Sentiti libero di rispondere con un esempio.

They mean the same thing. If you want to get VERY technical, recall that "won't" is the contraction of "will not." So, you are comparing:

She will not change her mind. vs. She is not going to change her mind.

Again, being very technical, "will not" indicates a refusal on the part of someone. It is an act of will. So, it is technically more correct to say "It is not going to rain tomorrow" than to say "It will not rain tomorrow."

In your case, since you are talking about a person refusing to do something, "will not" is the more correct phrase.

But, and I can't stress this enough, a native speaker is going to use these two phrases interchangeably.