-d, --diff, --compare
find difference between archive and file system.
Each option listed is an alternate way of saying the same thing. "-d" is equivalent to "--diff" and it's equivalent to "--compare". All of them mean the same thing, and you only need one of them on the command line.
Again, both formats say the same thing, but it's slightly different from above. This option requires
more information from the user (as evidenced by the "=DIR" portion). You can use either form "-c" or "--directory" but both must have a directory option. For "-c" the command line would look something like "tar -c directory_name". For "--directory" it would look something like "tar --directory=directory_name". Slightly different formatting, but again, the same meaning. Only one needs to be present on the command line.
EDIT: Beaten to the punch.... again... by Tink