Your question should be "What is the difference". Both sentences are incorrect. You can say either "I will saying nothing to him" or "I won't say anything to him". They both mean the same thing. Unless you mean "I will say something", saying "I won't say anything" is incorrect. It makes sense logically: two negatives make a positive, so the sentence literally means the opposite of what you're trying to say. "Anything" means "at least one thing", so "I won't say anything" makes sense both grammatically and logically. In some European languages it is grammatically correct to use a double negative to express a single negative. In a dialect of English in the US strongly influenced by Italian (I'm not sure exactly.where in the US this dialect is located) this is sometimes used, as well as in African American English. In both these cases, the use of double negatives is considered sub-standard.

Hi Camilla: In addition to Jmat's excellent answer, I would offer you some examples of several ways that you can express your idea. (a) I will not say anything. (b) I will not say anything to him. (c) I prefer not to speak to him. (d) I have no intention of speaking to him. (e) There will be nothing said between us. (f) I will offer him no answer. (g) I will offer him no answer at all. (h) I will have nothing to do with him. (I) I will offer no response. (j) I prefer to keep my distance from him. (k) I will avoid such a person. (l) I am very selective about who I will speak to. (m) I will not respond to a person like that.