Adding to @topomorto's great answer, let me add something. It is a very difficult question to answer in a few paragraphs.
First of all, '-ㄴ지 (-는지) 모른다 (안다)' is an idiomatic expression where the (conjunctional) connective suffix '-ㄴ지 (-는지)' is usually used with '알다 (to know)' or '모르다 (to not know)' to express your guess between the two choices. The reason why I used the word 'idiomatic' is the dictionary of National Institute of Korean Language has 31 meanings for '지', but unfortunately it doesn't list its meaning above-explained. According to Daum Korean dictionary, there are following exmaples which can help you understand it:
동사나 ‘있다’, ‘없다’의 어간 또는 선어말 어미 ‘-으시-’, ‘-었-’, ‘-겠-’의 뒤에 붙어, 주로 동사 ‘모르다’와
함께 쓰여, 추측을 나타내는 말.
나는 철수가 어디로 갔는지 모른다. I don't know where Cheol-su went.
봉호가 벌써 그 일을 끝냈는지 (안끝냈는지) 모르겠다. I don't know whether Bong-ho finished the work
You can change the above sentences using '안다' when you don't use 'whether' to translate in English, in other words, where there are no two choices. For example:
나는 철수가 어디로 갔는지 안다. I know where Cheol-su went. (In this sentence, there is no choice between A and B, but '어디로 갔는지 where he went' is the object of a transitive verb '안다'.)
This sentence is perfectly idiomatic. However,
?봉호가 벌써 그 일을 끝냈는지 안다. I know whether Bong-ho finished the work
It doesn't sound idiomatic. The perfectly idiomatic expression for the above sentence would be
봉호가 벌써 그 일을 끝낸 것을 (or 걸) 안다. I know (the fact) that Bong-ho finished the work. 'To know' is to know what happened factually. To know is not just guessing between A and B.
여자는 차가 어디에 있는지 몰라요. The object of 'not to know' is where the car is. The suffix is used to indicate the object of 'not to know' and she has to guess to know it. That's why it is more idiomatic than 여자는 차가 어디에 있는 걸 (것을) 몰라요.
아이들은 아기가 자는 걸 몰라요. The children don't know (the fact) that the baby is sleeping.
It is quite subtle difference and it is extremely difficult to explain in English. But, when you have two possible choices as in
아이들은 아기가 자는지 (안자는지) 몰라요. Children don't know whether the baby is sleeping or not (sleeping). Children has to guess to know it.
Let me take another example:
그녀가 예쁜지 (안예쁜지) 모른다. I don't know whether she is pretty or not. I have to guess.
차가 비싼지 (안비싼지) 모른다. I don't know whether the car is expensive or not. I have to guess.
Conclusion: I am not sure whether my answer adds any additional value to the other answer. But it boils down to (1) when you have to guess, you have to use '-ㄴ지' and (2) when you know the fact and don't have to guess, you have to use '-걸 / 것을'. You will need to read more example sentences to get to the bottom of the differences.
Isn't it correct that both express a conjunction that in English?
No. But very close.