I'm trying to get the difference between the verb write and the phrasal verb write out.
I wrote out my CV.
I wrote my CV.
As far as I'm concerned, they both have the same meaning.
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My answer will be for U.S. English.
Let's say I'm giving someone instructions:
I want you to write out a list of possible solutions to your problem.
This means I want you to take some ideas that are in your head, and go to the trouble of putting them down on paper. Implicit in this sentence is the assumption that if I am not quite explicit, the person will probably not bother writing anything down. I am imagining saying this to one of my children.
I can't imagine saying this in the context of creating a CV. For a CV, I suppose I might say
Please write up a draft of your CV and I'll help you format it.
The copying meaning mentioned in the comments sounds reasonable too, but in this day and age that doesn't seem very common to me.
We're dealing with the difference between a general verb (to write) and the phrasal verb formed with it (to write out). In short, write out distinguishes how one makes a copy or transcription of something or emphasizes the literal act of writing as distinguished from general composition. First I will describe the basic usage of to write out. Then I will show that UK and US differences, if they exist, are hard to track.
I. General meanings
A. Make a copy / write fully
Here is the first and pertinent meaning of to write out within "write, v." in the Oxford English Dictionary:
To make a (fair or perfect) transcription or written copy of (something, a rough draft, etc.); to copy out; also, to transcribe in full or detail, as from brief notes or shorthand.
to write out fair, to make a fair copy of.
Examples go back to Thomas Cooper's 1548 text Bibliotheca Eliotæ, where he defines the Latin verb describo:
Describo, psi, ere, to write out of a copy, to or∣der or appoynt, to paynt or wryte aduysedly, to declare or describe.
The emphasis is on the preposition out, which makes the action feel more extended or exhaustive. (The ex- words aren't a coincidence: ex- is often translated as out, such that exscribo literally means "to write out" or "copy." Similarly, extendo's non-Latinate translation would be "to stretch out.") Here are a couple of applicable meanings of the preposition "out" that may have influenced to write out, again from the OED.
First, out could refer to writing something out to the full extent, as in a copy:
4a. To the full extent, to a fuller extent (in space or time).
(examples) to draw out: to draw out 3 at draw v. Phrasal verbs, to draw out 4 at Phrasal verbs. to eke out: eke v. 3.
There is also the related sense of doing something to completion, as in writing a complete copy or writing something completely:
** In figurative uses expressing a change in state or progress towards a result.
6. a. To the conclusion or finish; to an end, to completion, to exhaustion.
... (b) In phrasal verbs with the sense ‘to bring to a conclusion by the action of the simple verb’. Frequently with it as indefinite object. Cf. to have out 2 at have v. Phrasal verbs.
10a has a similar sense of completion after difficult work:
10a. So as to reach a full result or solution (esp. by slow or laborious work).
Usually in set combinations with particular verbs: see figure v. 15c, to find out 4 at find v. Phrasal verbs, help v. 6c, to work out 7a at work v. Phrasal verbs 1.
All of these forms create phrasal verbs and they all predate to write out. In English, this phrasal construction for out is preferred to making out a prefix in these senses, which is why we have eke out instead of * outeke and write out (meaning to copy out) instead of * outwrite (which today means to write better than another).
Thus to write out attains a connotation of completion or extension that continues into the present sense of the word. Merriam-Webster and Cambridge Dictionaries use "full and complete form" and "all the necessary details" to explain out.:
M-W: to write especially in a full and complete form
CD: to write something on paper with all the necessary details
B. Literally write it
Finally, with 6b ("to bring to a conclusion by the action of the simple verb"), note the related sense of to write out that emphasizes the method of writing. In this sense, while one could write (generate text) by many methods, including keyboard, to write out would emphasize using the action of writing. Cambridge Dictionary hints at that ("to write something on paper") and American Heritage Dictionary documents expression in writing (presumably by hand) separate from the sense of generating a copy:
To express or compose in writing: write out a request.
To write in full or expanded form: All abbreviations are to be written out.
II. US and/or UK
Historically, the phrasal verb originated in England meaning to copy, transcribe, or write more fully. Dictionaries document this sense without denoting a single origin, and indeed this seems to be a phrase that is mutually intelligible and commonly used by both US and UK speakers. I will share results gathered in a recent US corpus (Corpus of Contemporary American English) and a similar UK Corpus (British National Corpus).
American examples (from 292 results in ~560 million word corpus):
Mo Daviau, Every Anxious Wave, 2017 - "When I return from the past, I sit alone with a notebook and write out the lyrics to songs I'd just heard performed."
"Once an illegal immigrant and a foster child ...", OregonLive.com, 2017 - "Pored over U.S. history lessons before taking her citizenship test, but says the interviewer asked her to do only one thing: Write out this sentence -- " I like to go to the store."
Ethan Chatagnier, "Miracle Fruit," Middlebury, 2016 - "I write out the document on a piece of stationery and place it like a bookmark in chapter six of Genesis in my mother's Bible."
British results (from 114 results in ~100 million word corpus)
Enid, interview, 1992: " So I came across these things you see when we were first married because I used to write out all Hector's bills for him, by hand before we had a typewriter."
Kevin, interview, 1991: "And Jan made them write out their names and some of them couldn't write their name!"
Pamela Scobie, A Twist of Fate, 1990: "She rubbed out the H with a single furious sweep and began to write out the entire alphabet in order, one impeccable letter after another."
I don't see a significant difference in meaning between the two corpuses, where usages overlap the sense of copying/writing fully/transcription with literally writing down. Thus I suspect that claims about differences in usage come down to finer distinctions in dialect or personal usage, or to specific contexts not reflected here.
A final touch of commonality: I'll even illustrate how one can write out the same document:
US: Colleen Coble, Mermaid Moon, 2017: "It took him forever to write out the check."
UK: Betty, interview, 1991: "I said to Bet I said they ought to write out a cheque on the end."
Receipts and bills are also commonly written out on both sides of the pond, at least when those processes are not automated.