A differential backup will backup changes since the first full backup.
You didn't enough of Section 5.2 of the tar info manual. Just after the instructions to create an incremental backup (the first incremental backup), it explains how to create a subsequent level.1 (differential) backup:
The created archive `archive.2.tar' will contain only these three
members. This archive is called a "level 1 backup". Notice that
`/var/log/usr.snar' will be updated with the new data, so if you plan
to create more `level 1' backups, it is necessary to create a working
copy of the snapshot file before running `tar'. The above example will
then be modified as follows:
$ cp /var/log/usr.snar /var/log/usr.snar-1
$ tar --create \
Just keep copying the initial snapshot file and use the copy for the -g (--listed-incremental) argument. Then the files modified from the initial backup will be backed up because you are using the snapshot file from the initial full backup.