They are synonyms. We use 'idiot' and 'fool' to desribe a person who does not behave or speak in a reasonable or clever way.

Personally, I see "idiot" as more mean, and "fool" as more general and sort of used in a joking way. But that's just my opinion. By definition alone, they are basically interchangeable. [Examples of how I feel the "tone" of the words works: Person A: I really think grass is blue. Just look at it, it's blue! Person B: Wow, you are truly an idiot. Person A: Wait, I forgot where I live... Person B: Haha, you fool.]