Natural Language: Understood by humans and follows certain grammar but flexible to change within different dialects like there could be sentence variation in english to describe particular event as region change, Ex: English and Gujarati. LLMs can interpret and generate these languages after learning and adjusting weights in a neural network. You can not write Turing Machine for this language.
Formal Language: A formal language is an abstract structure, but a natural language is not, because its rules of grammar and syntax are open to debate and interpretation. While natural languages as a whole are generally considered to be open to interpretation and have flexible grammar and syntax, specific dialects or registers within a natural language family can exhibit more formal characteristics. For example, certain styles of written English, such as academic or legal writing, may adhere to stricter rules and conventions, resembling formal language in some aspects while still being part of the broader natural language category. You can not write Turing Machine for this language.