This is the best I could come up with:
git log --name-only --pretty=oneline --full-index HEAD^^..HEAD | grep -vE '^[0-9a-f]{40} ' | sort | uniq
Replace HEAD^^ and HEAD with the commits you want to compare.
My attempt uses git log with --name-only to list all files of each commit between the specified ones. --pretty=oneline makes the part above the file listing consist only of the commit SHA and message title. --full-index makes the SHA be the full 40 characters. grep filters out anything looking like a SHA followed by a space. Unless you have files beginning with a SHA followed by a space, the result should be accurate.