"Writing things down" vs. "writing things up"

Is there any difference in the usage of "writing things down" vs. "writing things up"? Are they both correct?

"Write down" is an idiom which means "make a note of", or "get something written".

"Write up" is an idiom which means "write an account, record or essay" about something.

Both are correct. It depends on the circumstances. I would "write things down" when the purpose was to help remember them- "I need to write things down or I will forget them in a few days." I would "write things up" when the purpose was to summarize for distribution or publication- "I'll write up the notes from the meeting and send them out this afternoon." Of course there is also 'written up' which usually implies that some sort of infraction was incurred of which written note was taken- "I brought a plastic knife to school in my lunch and was written up by my principal."

Generally I'd use "writing things down" to refer to writing fairly short things, e.g. taking notes, where the emphasis is purely on making sure that things are in writing. Example - a teacher might say to their students "make sure you're writing this down".

On the other hand, I'd use "writing things up" to mean writing something longer - an article or an essay about an event/idea/proposal/etc. Example - two colleagues reach the end of a discussion about something, and one asks the other "could you write that up and email it to the team?"

My usage may not be universal, but to me, "writing things down" emphasizes what is written, and "writing up" emphasizes what is written about. For example, I might write up (or do a write-up on) an event by writing down a list of people who attended.