According to older grammar, one should say "I shall" when referring to the future: "I shall visit you next week." One should use "I will" when expressing determination: "I do not like him. So I won't talk to him if we accidentally meet at the party." Here in the United States in 2021, everyone uses "won't" for the future, too. If you say "shan't," people will look at you funny. They might think that you are joking. Personally, I like "shan't" because I am very old. (I shan't tell you my age.)

Yes there is, shan't is a very old english way of say shouldn't; where as won't sounds a bit more modern.

Hello! Both mean the same but shall / shall not is almost never used in English and I have even barely seen it in academic or literary texts because it is such a rare word. I would just stick with using will / will not because it's the most commonly used, native speakers don't usually use shall / shall not, we could even say it's not used anymore almost under any circumstances & I've never heard a native speaker say it. I hope to have helped you!

There's a difference in formality I guess you could say, "Shall not" which "Shan't" is a short form of is a very formal way of saying something, "shan't" is not a phrase used often, or at all to be honest with you