diff to output only the file names

You can also use rsync

rsync -rv --size-only --dry-run /my/source/ /my/dest/ > diff.out

From the diff man page:

-q   Report only whether the files differ, not the details of the differences.
-r   When comparing directories, recursively compare any subdirectories found.

Example command:

diff -qr dir1 dir2

Example output (depends on locale):

$ ls dir1 dir2
dir1:
same-file  different  only-1

dir2:
same-file  different  only-2
$ diff -qr dir1 dir2
Files dir1/different and dir2/different differ
Only in dir1: only-1
Only in dir2: only-2