What is the difference between remove and completely remove in synaptic package manager?

To add a little bit to what Rinzwind and nickguletskii said, completely remove will remove configuration files in /etc directory, but never touch personal configuration files in your home folder, that can only be removed by hand.

The 'complete removal' option instructs Synaptic to remove any configuration files associated with the package as well.

Source: community wiki

Normal removal is equivalent to sudo apt-get remove application using terminal.

Complete removal is equivalent to sudo apt-get remove --purge application.

Files in your home folder are not part of the package so they will stay on your system until you remove them yourself.

Removal will remove the package. Complete removal will remove the package and also remove configurations associated with the package.

So if you remove and then install a package, your settings will still be there. If you completely remove and install a package, your settings will be kept.

However, this will only remove configurations in the /etc directory, so that no personal information gets lost. Thanks to @enzotib for the clarification!