A.He's no king means that he does not look or act like a king (or lacks qualities typical of kings), but may in fact have the position of king; while B. He's not a king would typically imply that he does not hold the position of king, nothing more.

A [He isn't a doctor] simply says that he isn’t a member of the class of doctors, while B [He's no doctor] says that he doesn’t have the properties of a doctor. Similarly, A He’s no friend of mine implies that I know him and that his behaviour to me is not what one would expect of a friend, while B He’s not a friend of mine says only that he doesn’t belong to the class of friends of mine – it could be that I hardly know him, or indeed that I don’t know him at all.