The man page is many times better than me at explaining it:

       upgrade
           upgrade is used to install the newest versions of all packages
           currently installed on the system from the sources enumerated in
           /etc/apt/sources.list. Packages currently installed with new
           versions available are retrieved and upgraded; under no
           circumstances are currently installed packages removed, or packages
           not already installed retrieved and installed. New versions of
           currently installed packages that cannot be upgraded without
           changing the install status of another package will be left at
           their current version. An update must be performed first so that
           apt-get knows that new versions of packages are available.

       dist-upgrade
           dist-upgrade in addition to performing the function of upgrade,
           also intelligently handles changing dependencies with new versions
           of packages; apt-get has a "smart" conflict resolution system, and
           it will attempt to upgrade the most important packages at the
           expense of less important ones if necessary. The dist-upgrade
           command may therefore remove some packages. The
           /etc/apt/sources.list file contains a list of locations from which
           to retrieve desired package files. See also apt_preferences(5) for
           a mechanism for overriding the general settings for individual
           packages.