¿Cuál es la diferencia entre "to write down" y "to write" ?Puedes dar oraciones como ejemplo.

"to write down" has the meaning "to put on paper". You use this when you are writing something that currently doesn't exist on paper for the purpose of having it on paper for later reference.

For example,
"Let me write down my phone number for you."
"Write down your name and address here."
"He wrote down all the people's names."

"to write" is more simple and general.

"to write" can be used pretty much anywhere "to write down" is used, but not the other way around.

We don't say
"I wrote down a letter to my sister." because you are coming up with the letter's contents as you write. The letter contents do not already exist for you to note.

You can think of "to write down" as "noter" in French, whereas "to write" is more simply "écrire".