They might both be correct, but I think "The cat is in the box" might be the better choice. If someone asked "Where is the cat?" It would sound more natural to answer "고양이가 상자 안에 있습니다 (The cat is in the box)." If someone asked "What's in the box?" You would more naturally answer "상자 안에 고양이가 있습니다 (There is a cat in the box)." In this case, even though the wording is identical, the ordering of the words makes it more likely to be understood as "The cat is in the box."
I thought when you used ~가 is was the subject marker, which would make it "the cat" and ~는 is the topic marker so thats when there would be "there is".
What's the difference between "inside" and "in the"; instead of saying "...in the box", "...inside the box", is it the same?
Nevermind. I just realised that 아네 is inside. I can't delete my comment in the app now. Back then I could...