In the specific example used in the question, either copy or copy out could be used.
However, the use of copy out has a specific meaning:
: to write down (words that one is hearing or reading)
// Are you copying all of this down?
// I copied out the equations on a piece of paper.
In short, when you copy out something, you generally copy it in real time (simultaneously). You are copying it while listening to it or reading it.
On the other hand, to simply copy something (without out or down) can be used to create a duplicate of something without regard to time. Although, generally, to do something like copy a book would involve a series of time-separated copying-down sessions.
So, this would be fine:
✔ I copied the book.
But these would sound strange, mainly because a book is so long that it would not be a quick or one-session task:
✘ I copied out the book.
✘ I copied down the book.
To clarify the distinction even further:
✔ I copied the book by copying each page.
✔ I copied the book by copying out each page.
✘ I copied out the book by copying out each page.
✘ I copied out the book by copying each page.
To add to this, the verb also makes a difference. You could say the following (the first being more common):
✔ I wrote down what I'd heard earlier in the day.
✔ I wrote out what I'd heard earlier in the day.
But you would generally not say any of the following:
✘ I wrote what I'd heard earlier in the day.
✘ I copied out what I'd heard earlier in the day.
✘ I copied down what I'd heard earlier in the day.
✘ I copied what I'd heard earlier in the day.
Those aren't strictly wrong, but they're uncommon.