"Написать" doesn't put any stress on recording the information (for future use?), while "write down" does afaik. Its counterpart in Russian is "записать".
Thanks for that , but I still think that when someone is being asked to record in writing some particular words, we would (in the UK) indeed use 'write down'.
But would you use it if, say, instructing an elementary student to repeatedly copy a word to work on their handwriting? To me, "write down" implies the information is being recorded, while "write" in the imperative stresses only the physical action of writing, regardless of any content.
In that context you would probably say "write this word out five times". The duolingo English version suggests that you can do this any time you like anywhere but maybe this is what the Russian conveys?
1st is to understand 2nd is to learn 3rd is to correct 4th is to remember 5th is to know