I agree 100% with Brian. To add to his explanation, maybe think of "Put down" as an order, or to lay (smth) down/place (smth) down. You can put anything down. A pencil, a phone, even a person! As a side note, we also use the compound word 'put-down' (Meaning an insult... you're putting someone down).
- Put down that phone and pay attention to me!
'Write down' is probably the safest to use if your meaning is to take a pen/pencil and write something down.
'Take down' is less of an order than 'put down' and it's used slightly different.
- Take down those ugly curtains!
- Take down my number (so you can call me)
Take down can ALSO be used as a one-word noun: 'takedown' The police and FBI often have takedowns, where the bad guys might be wrestled to the ground with guns in their faces. (Isn't English fun!!!)