Put down - to write something, to make a note of something to write down - to write something on paper, especially in order to remember or record it. Разницы нет никакой.

If you put down something, you are generally holding something (an object) that you physically move or put down or place on top of something. For example, 'I put down the vase' means I put it down on a table or something. There is a slang type of phrase as well which means to denigrate someone. For example, if someone says something nasty about one of your friends, you might say "that was a bit of a put down" and mean that was a bit f a nasty comment. Write down is exactly that, to write something down in a notebook or on paper. You can also say 'I will put that down in my notebook' and mean I will write that in my notebook - just to confuse things a little :-)