Using cp is a good way to do it, but now that the files are copied, using the update flag may be a good way.
nohup cp -Rpu /original/directory /new/directory
R for recursive(it gets the entire directory)
p for preserve permissions, ownership and timestamps.
u for update(only copy the files that are newer, different size or does not exist in the destination directory).
Then everything should be the same throughout both directories.
Remember, if you run: cp /dir /dir2 then the directory dir will be copied INTO dir2, unless dir2 does not exist.
The easiest way to get what is in dir into dir2, would be to copy the contents:
cp /dir/* /dir2/
I've gotten into the habit of adding the trailing slash so that I know what the command was meant to do, even if I come back 3 weeks and look at history, seeing the trailing slash, I can tell that the directory did already exist, because that's my habit. If the directory already exists, I add a trailing slash, otherwise, I do not.