What's the difference between 너 못하나? and 너 못한가?

ANSWER: 너 못하나? means Can't you do it? and 너 못한가? means Are you inferior?

못하다 can be used as a transitive verb or as an adjective. When it is used as a transitive verb, it means cannot, be impossible, or be unable to. When it is used as an adjective, it means be inferior or be worse than.

Okay. But how can you know 못하나 is being used as a verb and 못한가 as an adjective?

Because the question endings -나 and -는가 are used only with verbs, and -ㄴ/은가 only with adjectives, though -ㄴ가 is also used with the copula (이다). Therefore, when you see 못하나 or 못하는가, you know 못하다 is being used as a verb; and when you see 못한가, you know it is being used as an adjective.

However, these days many Koreans are forgetting these distinctions and are mistakenly using the question ending -나 with both verbs and adjectives, so you have to watch out for that.