Difference between save and export

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As I described in my last post (http://tuhrig.de/difference-between-save-and-export-in-docker), there are two ways to persist a Docker images or container:

A Docker image can be saved to a tarball and loaded back again. This will preserve the history of the image.

# save the image to a tarball
docker save <IMAGE NAME> > /home/save.tar
 
# load it back
docker load < /home/save.tar
A Docker container can be exported to a tarball and imported back again. This will not preserve the history of the container.


# export the container to a tarball
docker export <CONTAINER ID> > /home/export.tar
 
# import it back
cat /home/export.tar | docker import - some-name:latest

Hi. I want to move a container with some new files from one docker host to another docker host. Should I use save or export? Please clear this confusion for me!