What's the difference between "неважно" and "ничего"

Question

What's the difference between "неважно" and "ничего"?
As I understand both mean "never mind", but what about usage?

for example when I ask somebody but he/she can't understand me, can I use both of them as the response to interrupt the conversation?

Answer 1

Ничего is not translated to never mind, at least in the context you provided.

In your example,

When I ask somebody but he/she can't understand me

given two choices you reply with Неважно if the question was not really important. Alternatives to that can be Проехали or Забудь.

Let's take a look at another example: someone accidentally steps on your shoes. If the person excuses, you would reply with:

• Never mind.
• No problem.
• That's okay.

In Russian you would reply with:

• Ничего [страшного].
• Все хорошо/в порядке.

Answer 2

Context 1: the difference is not essential.

• Что ты имел в виду? - Неважно.(- Забудь.)

• What did you mean? - Never mind. (- Forget it.)

• Что ты имел в виду? - Ничего.

• What did you mean? - Nothing (special).

Context 2: the difference is dramatic (bad vs OK).

• Как ты себя чувствуешь? - Неважно. (~ Плохо.)

• How do you feel? - I don't feel good.

• Как ты себя чувствуешь? - Ничего. (~ Неплохо.)

• How do you feel? - I'm OK.