write vs write down

write and write down both are verbs.

Word	Noun	Adjective	Verb	Adverb
write	No	No	Yes	No
write down	No	No	Yes	No

As verbs, write down is a hyponym of write; that is, write down is a word with a more specific, narrower meaning than write:

write (verb)	write down (verb)
communicate or express by writing	reduce the estimated value of something
record data on a computer	put down in writing; of texts, musical compositions, etc.
contact in writing
produce a literary work
mark or trace words or symbols on a surface
write or name the letters that comprise the conventionally accepted form of (a word or part of a word)
write music
create code, write a computer program