They are not interchangeable. "Take down" is more specific than "write down" .
You take down notes or information that you hear from someone else. Typically, a secretary might take down a letter that a boss dictates, or a journalist might take down answers that an interviewee gives. In other words, the writer takes the information from another source, and then writes it down.
If you're talking about the writer's own ideas, you CANNOT use "take down". For example, you might write down your thoughts and reflections in your diary at the end of the day - not "take down".