I watched a movie, the character said
You are no king.
I am thinking, why not say
You are not king,
You are not a king.
What's the difference?
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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 He's not a king would typically imply that he does not hold the position of king, nothing more.
[He isn't a doctor] simply says that he isn’t a member of the class of doctors, while [He's no doctor] says that he doesn’t have the properties of a doctor. Similarly, 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 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. (The Cambridge Grammar of the English Language p390)
The assertion "you are not [a] king" merely denies kingship, something that can be rectified by an external act such as a coronation. The "a king" variant is arguably the more dismissive of the two.
"You are no king" is a negation of "you are king", where the lack of an article before "king" is understood as a null-article. It references an intrinsic quality (kingliness) regardless of external factors such as a valid coronation.
If someone had been crowned king but consistently acted in a manner unbefitting of a king, "you are not [a] king" would be an invalid assertion, while "you are no king" might be justifiable.
No can be used as an adjective. As suggested in the M-W dictionary—pay attention to entry 2 in the adjective section—no as an adjective can mean "not a; quite other than a." The example given in that definition
He's no expert
is similar to your sentence "You are no king."
And so, your sentence basically means that "you" are not a king, or quite other than a king.
As for your other 2 sentences
You are not king.
You are not a king.
These sentences lack the emphasis in the word no that describes "you" as something quite other than a king, but just simply negate their position as a king.
And as for the difference between king and a king, king is more specific than a king. Using a king can just mean that "you" are not any king, but using king could be referring to a specific king position. See these examples:
Louis XIX was not a king. (He did not become king at all)
Louis XIV was not king of England. (He may not have been that king, but he was king of France.)