Hm. The difference is subtle and I would say they're generally interchangeable but 'note down' to me implies that the note taker is listening or paying attention to what's being said or whatever is happening. 'Write down' feels more like making exact copies of what's on the board for example, or what a teacher has just said, as opposed to taking notes for later reference. There are other instances where these are very different from each other, but I've used one example.