What is "dist-upgrade" and why does it upgrade more than "upgrade"?

From apt-get manual:

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. So, dist-upgrade
   command may 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.

And with the newer apt tool available from 14.04 onwards:

full-upgrade
   full-upgrade performs the function of upgrade but may also remove
   installed packages if that is required in order to resolve a
   package conflict.

In your particular case, I see, for example, that linux-headers is a virtual package that is provided by both linux-headers-3.0.0-12 and linux-headers-3.0.0-13 and that sounds like the kind of package installation and removal handled by dist-upgrade, but not by upgrade.

apt-get upgrade is restricted to the case where packages are to be replaced by newer versions, but no package needs to be added or removed. A new version of Firefox, for instance, should be installable with apt-get upgrade.

However apt-get upgrade will refuse to work when there are additions or removals required by the updated versions. For example, when you have kernel linux-image-3.2.0-10-generic installed and linux-image-3.2.0-11-generic appears, the linux-image-generic package gets updated to depend on the newer version. In order to install the new kernel, you need to run apt-get dist-upgrade.

Notice how an apt-get upgrade will say that the kernel packages have been held back. That's the cue for using apt-get dist-upgrade.

Basically upgrade will only upgrade existing packages from one version to another. It will not install or remove packages, even if doing so is required to upgrade others. In the case of kernel updates, upgrading the linux-generic package requires installing the new linux-3.0.0-13-generic package, and since upgrade refuses to install or remove packages, it refuses to upgrade linux-generic.

Sometimes various incompatibilities between packages will require some packages to be removed in order to upgrade others, and that will also require dist-upgrade. Kernel updates will always require dist-upgrade because of how they are handled. Rather than have a kernel package that gets updated, an entirely new kernel package is created each time, and the kernel metapackage is updated to depend on the new kernel package instead of the old. This was done so that you keep the old kernel versions around so that in case there is a problem with booting the new kernel, you can choose the old one from the boot menu and recover.

The apt-get upgrade command will normally only install updates (or fixes) to currently installed packages. Typically a new release of Mozilla Firefox, for instance, would be installed with this command.

However apt-get upgrade will NOT generally install new releases, where major changes (including removal of packages or GRUB update is required). For example, when a new Linux kernel (linux-image-3.x.x-xx-generic, etc.) is available, the package will not get installed.

In order to install the new kernel, you will need to run apt-get dist-upgrade. You will be notified when you run apt-get upgrade, as it will say that certain packages have been held back. That's your cue to use: apt-get dist-upgrade.

Your best option is:

apt full-upgrade

-which auto-handles the dependencies when upgrading packages; whereas:

apt upgrade

-does NOT auto-handle dependencies when upgrading packages.

Short answer those who want the exact point

apt-get update

• The update command updates the package list with the latest available versions
• It won't upgrade the installed packages just update the package list with a new version
• To upgrade need to run the upgrade command

apt-get upgrade

• To update the already installed packages to the latest available version
• It will upgrade only the installed packages that already available to the latest version version
• If the dependencies are missed the current version will be kept without any upgrade

apt-get dist-upgrade

• To update the already installed packages to the latest available version with required dependencies
• It will automatically install the required additional packages and dependencies and also removes the already installed packages to make the clear upgrade

apt-get full-upgrade

• full-upgrade performs the upgrade function but may also remove installed packages if required to resolve a package conflict.

apt-get do-release-upgrade

• It can upgrade the release version of ubuntu for example if you want to upgrade ubuntu 18.04 to 20.04 we can run do-release-upgrade
• To use this command, the system must first be fully upgraded. To do that, you should first run sudo apt-get upgrade, followed by sudo apt-get dist-upgrade

ref:https://linuxhint.com/apt_get_upgrade_dist_upgrade/#:~:text=Unlike%20apt%2Dget%20upgrade%20command,sudo%20apt%2Dget%20dist%2Dupgrade