I'd use "write it down" in a situation like this: "Here is the phone number. Please write it down."
"write down" emanates a sense of urgency or implies that something is actually put down on a piece of paper.
"write": He used to write short stories. (create something rather than just write words on paper)
So we use "write down" when it's important to transfer words onto paper. We can use "write" in that context too, but it's also often used in a broader sense.
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