What's the difference between the apt packages: 'cinnamon' and 'cinnamon-desktop-environment'?
Look at the output of:
apt show cinnamon cinnamon-desktop-environment cinnamon-core
Basically you can say that cinnamon is a part of cinnamon-core which itself is a part of cinnamon-desktop-environment.
cinnamonis the most basic and minimal Cinnamon installation you can get.cinnamon-coreis a minimal Cinnamon desktop, but it also includes few more packages thancinnamon, like the e.g. file managernemo.cinnamon-desktop-environmentis a full desktop environment suite which also includes standard applications like a browser, mail client, document viewer, editor, picture viewer, multimedia viewer, etc...
While I cannot answer what is not included in cinnamon as opposed to cinnamon-destop-environment I can answer your broader question.
cinnamon-destop-enviroment is a meta package. A meta package does not include any real programs in it it is more of a recipe. It directs the package manager to pull in all the needed programs for the desktop environment to work fully and have all the features it should. It will pull in cinnamon and all the other files needed for a full Cinnamon Desktop install
Without it, just installing cinnamon would most likely get you a working desktop but features would be missing. For example you could be missing the bluetooth manager for Cinnamon. As some of this stuff would be installed in standard Ubuntu, you would likely have a functioning desktop but might have to do some stuff in terminal only because you are missign the graphical front-end in Cinnamon. I will also make it much harder for you to get help because instruction for Cinnamon will not work because you are missing a program needed in the instructions.
In short you want to install the cinnamon-desktop-enviroment to get the full desktop. In doing so, however, you will have doubles of some programs. Nautilus will still be installed but Cinnamons default Nemo will also be installed, so if you search for "files" you will come up with two programs with that name, you will need to search for Nemo.
One other word of warning, you may find it harder to get help here with Cinnamon installed, it is a minority DE so less people will be available to help you. If you are just trying to get a Gnome2 like desktop, you might look at installing ubuntu-gnome-desktop and using the Gnome Classic desktop from the login screen, it looks and works like the old Gnome2 desktops.
Installing the ubuntu-gnome-desktop will affect your Ubuntu Unity, especially the top bar. Everything will still work in Unity but the icon will be a mix of Unity's default and Gnome's default icons. Ubuntu Gnome in a official flavor of Ubuntu so is supported on these forums. As you have a fresh install you could just reinstall Ubuntu Gnome instead of installing ubuntu-gnome-desktop but the result will be the same.
There is one bug with Ubuntu Gnome that affect Nautilus, it causes freezing in the menu. If you want to use Gnome, you could install another file manager, the closest would be Nemo(my preference, it's Nautilus with more features). To do that there are a few step but they are easy, see http://www.webupd8.org/2013/10/install-nemo-with-unity-patches-and.html for instructions.