Always use rsync for copying files, because It Is Great.
To ignore existing files:
rsync --ignore-existing --recursive /src /dst
Do read the manual and search around for many, many great examples. Especially the combination with ssh makes rsync a great tool for slow and unreliable connections on account of its --partial option. Add --verbose to see which files are being copied. Be sure to check out the plethora of options concerning preservation of permissions, users and timestamps, too.