Difference between Nothing and Anything

Last Monday a partner asked to the proffesor which is the difference between nothing and anything, but at that moment he was explaining something else, so I think he forgot it, and as I know the answer, I would like to share it with you with some sentences as examples:

● I see nothing./ I do not see anything.

● She is doing nothing./ She is not doing anything.

● We have nothing to do./ We do not have anything to do.

● Tell me nothing./ Do not tell me anything.

As you can see in the examples, we do not use "not" when we use nothing because that word is already negative, anything is also negative, but it is mandatory to use "not" with it. 

I hope my little explanation was helpful for all of you.