The difference between these words is in the way of usage of them: in English we can't have double negative in the same sentence, so for example: we will say nothing, I know nothing about it, but with anything we can use not, so, we won't say anything, I don't know anything. And also, anything can be used in an affirmative for: If anyhing happens, just let me know.

They are opposites. Anything is used for a thing, including all things. Nothing, the opposite, is no thing, or not including a thing at all. It is like cualquier cosa y nada. I hope this helps!