"Having nothing to say" can literally mean you have nothing to say. But this phrase can also be said like "I have nothing to say to you" which is something you'd say to someone you're angry with and don't want to talk to. "Having nothing to do with" basically means it's irrelevant. "He has nothing to do with me" means we don't have any business together or do anything together.

HAVING NOTHING TO SAY TO YOU- This means you literally have no more words to say to someone. You use this expression when you are upset with someone or perhaps shocked about what you have just heard. HAVING NOTHING TO DO WITH YOU - This means you have had a fall out with someone. You are no longer going to speak or deal with someone. This can occur with a friend or in a business situation, " I am having nothing to do with your company" means I am not dealing with your company anymore.