Is updating via Update Manager equivalent to updating with apt-get?

Was my desktop activity equivalent to my server activity

Most of the time, yes, the desktop update is equal to apt-get update && apt-get upgrade.

However, apt-get upgrade will not install packages like kernel upgrades that are optional but highly recommended, while Update Manager will. You have to use apt-get dist-upgrade for that, in which case you would have a true equivalent.

If you use apt-get upgrade and kernel upgrades, etc. are available, you will see a message like The following packages have been held back:; that's your cue to run apt-get dist-upgrade if you wish to upgrade those packages.

Note that a number of types of cloud servers do not use their own kernel, but their physical "host" machine's kernel (OpenVZ, vserver, LXC, Xen paravirtualized); other types behave like physical machines themselves (Xen HVM, KVM).

For more light into Update commands refer to the AptGet How-to:

Maintenance commands

1. apt-get update - Run this command after changing /etc/apt/sources.list or /etc/apt/preferences. For information regarding /etc/apt/preferences, see the Pinning How-to. Run this command periodically to make sure your source list is up to date. This is the equivalent of "Reload" in Synaptic or "Fetch updates" in Adept.

2. apt-get upgrade - This command upgrades all installed packages. This is the equivalent of "Mark all upgrades" in Synaptic.

3. apt-get dist-upgrade - The same as the above, except that it adds the "smart upgrade" checkbox. It tells APT to use "smart" conflict resolution system, and it will attempt to upgrade the most important packages at the expense of less important ones if necessary.

Note: apt-get dist-upgrade does not perform a distribution upgrade. See the upgrading page for more information.