I think you can try using
wget --no-clobber, but as mentioned above, you probably want to look into using a solution that is based on
rsync rather than HTTP. Presuming that you have SSH access to the server,
rsync can use that as a transport mechanism with
rsync -za --stats -essh firstname.lastname@example.org:/path/to/files /path/to/local/copy/of/files.
Note though that
wget --no-clobber -r will only get files that are new since the last check, and won't re-download new copies that have changed. That's why
rsync is the better solution for the use-case you present.
Another alternative I found is the GPL software, HTTrack, which mirrors entire web sites, and can pull down subsequent differentials. It can be found here. Windows screen shot, but there are builds and/or source for Windows, OS X, Linux, BSD, and Android(!).