write vs jot down

write and jot down both are verbs.

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

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

write (verb)	jot down (verb)
communicate or express by writing	write briefly or hurriedly; write a short note of
record data on a computer
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