apt-get are 2 different ways of installing software. Basically apt-get, aptitude, and synaptic are built on top of debian's dpkg package management program. They all perform the same basic function - package management, but have some extra features. 1 of the extra features of apt-get is that it will install dependencies and dpkg does not.
Regarding the -p/purge ...
--purge and will remove everything, including setttings and configuration files. From the manual:
-r, --remove, -P, --purge package...|-a|--pending
Remove an installed package. -r or --remove remove everything
except conffiles. This may avoid having to reconfigure the pack‐
age if it is reinstalled later. (Conffiles are configuration
files that are listed in the DEBIAN/conffiles control file). -P
or --purge removes everything, including conffiles. If -a or
--pending is given instead of a package name, then all packages
unpacked, but marked to be removed or purged in file
/var/lib/dpkg/status, are removed or purged, respectively. Note:
some configuration files might be unknown to dpkg because they
are created and handled separately through the configuration
scripts. In that case, dpkg won't remove them by itself, but the
package's postrm script (which is called by dpkg), has to take
care of their removal during purge. Of course, this only applies
to files in system directories, not configuration files written
to individual users' home directories.
Removing of a package consists of the following steps:
1. Run prerm script
2. Remove the installed files
3. Run postrm script
The same goes for
remove is identical to install except that packages are removed
instead of installed. Note the removing a package leaves its
configuration files in system. If a plus sign is appended to the
package name (with no intervening space), the identified package
will be installed instead of removed.
purge is identical to remove except that packages are removed and
purged (any configuration files are deleted too).
Basically it is the same option. Mind you: removal of dependencies does not happen with dpkg. apt-get does remove dependencies
Documentation from comment by Lekensteyn: