Export only modified and added files with folder structure in Git

git diff-tree -r --no-commit-id --name-only --diff-filter=ACMRT $commit_id

1. git diff-tree -r $commit_id:

   Take a diff of the given commit to its parent(s) (including all subdirectories, not just the top directory).

2. --no-commit-id --name-only:

   Do not output the commit SHA1. Output only the names of the affected files instead of a full diff.

3. --diff-filter=ACMRT:

   Only show files added, copied, modified, renamed or that had their type changed (eg. file → symlink) in this commit. This leaves out deleted files.

UPDATE FROM THE COMMENT:
Base on the question context and the comments below, with the following command, you can get the ACMRT files as a .tar file with their folder structure.

git diff-tree -r --no-commit-id --name-only --diff-filter=ACMRT $commit_id | tar -czf file.tgz -T -

git diff-tree -r --no-commit-id --name-only --diff-filter=ACMRT $commit_id | xargs tar -rf mytarfile.tar

Just to round this out, here is the command piped to tar. This exports the files into a tar archive.

Here is a one-line command that works on Windows 7. Run it from your repository's top-level folder.

for /f "usebackq tokens=*" %A in (`git diff-tree -r --no-commit-id --name-only --diff-filter=ACMRT HEAD~1 HEAD`) do echo FA|xcopy "%~fA" "C:\git_changed_files\%A"

• echo FA answers the inevitable xcopy question about whether you're copying a file or a directory (file), and the possible question about overwriting a file (overwrite All)
• usebackq allows us to use the output from our git command as the input to our do clause
• HEAD~1 HEAD gets all differences between the previous commit and the current HEAD
• %~fA transforms the git output into fully qualified paths (needed to change forward slashes to backslashes)
• C:\git_changed_files\ is where you will find all the files that are different