Updating rsync through Homebrew vs updating through .tar download

When you install rsync through HomeBrew an additional patch with a bugfix is applied to the code before it is compiled and installed on your system. You will not (automatically) get this patch when installing manually from a .tar.gz.

HomeBrew also ensures to install dependencies easily. You will have to do that manually when doing it yourself.

Finally HomeBrew makes it easier to uninstall rsync again, if you do not need it anymore.

The software offered on rsync.samba.org and installed in the guide you link is almost identical to the software installed through HomeBrew, except for the extra patch I mentioned.

The software on GitHub you link to is actually not rsync. It is a graphical user interface for rsync, that requires you to have rsync installed in advance.