Which way is correct here? Some explanation would be appreciated.
"Company name" is a noun phrase that is used on legal papers and other forms when dealing with a company as an artificial person or legal entity. Typically in these cases, "your" would not precede "company name" because the legally company represents itself.
This phrase is used in correspondence between the company and another legal entity.
"Your company's name" is an informal phrase which means "the name of the company that you own (in part or in full)" or possibly "the name of the company that employees you".
This phrase is used in conversations among real people.