First off, I want to preface my thoughts by saying that they are my opinion. Some people will agree with it. Some will not agree. That's what makes literature fun. It's open to interpretation. Math isn't, which is also why a lot of people like math too. Secondly, based on the picture, it would seem that your teacher wants to hear your opinion. There isn't a right/wrong answer in this case. State your opinion, and clearly explain why you believe that.
I can't read the entire question in the picture, so I'll go with your exact wording. Based on that question, I think the "big idea" of the book is the "main idea." Here's why. I don't have any huge literary reason other than I am taking "big" and "main" as synonyms of each other. I coach soccer in addition to teaching, and I know that I have used those two words interchangeably. "Number 11 is the main threat from their team." "Number 11 is the big threat from their team." Both sentences mean nearly the exact same thing. You could argue there are some differences, but my athletes know exactly my meaning regardless of which word I used.
If you are reading a book, and the author's main idea is to inform you about what it takes to survive in the wild, then that is also the book's "big idea." To teach you about what it takes to survive in the wild.
Now here's where I'm going to sound a lot like an English teacher (because I am one). Book themes are big ideas. That's why they are themes. Let's use "Star Wars" for an example (the ones with Luke Skywalker). You might say that the main idea is the struggle between the Empire and the Rebels. You are totally correct, because without that struggle there wouldn't be much of a plot. Thematically though, there is a focus on good/evil (light/dark). That theme runs through the entire series, and I don't think anybody would argue that theme isn't a big idea for the "Star Wars" franchise. Another thematic big idea is courage and heroism. Plus the importance of family and relying on faith. Those are all super important parts of the films. They are "big ideas" that help support and deepen the main BIG IDEA of a struggle between the Empire and Rebels.
A book can have a number of different themes or overarching subjects which are present throughout the book. However, a book generally only has one main idea; therefore, the main idea is the big idea of the book. The main idea can be characterized as the most significant, or most detailed, of the themes that are covered in the book. Although different themes such as racism, poverty, and crime can be present in a novel, the main idea generally focuses on how these themes interplay to create the circumstances that drive the story along. Thus, the main idea of a book tends to be more detailed than the general themes of the book.
The "big idea" is the overall main theme. For example, in "How to Kill a Mockingbird" the big idea is racial and social tension. As writer15 said, there can be different themes or different subject matter; however, themes and subject matter revolve around a central idea.