I understand MATE is a desktop environment, but what is the difference between ubuntu with a MATE desktop environment and ubuntu-mate ?
EDIT: Difference Between Ubuntu and Ubuntu-Gnome is not a duplicate of my question because there is a dedicated distribution called ubuntu-mate
What is MATE DE?
quoting from the MATE Wikipedia article
But what is a Desktop Environment?
quoting from the Desktop Environment Wikipedia article
In Linux there are many different DEs. The most well-known are: Unity, GNOME 3, KDE, Cinnamon, XFCE, and LXDE.
So what is Ubuntu MATE?
quoting from the official Ubuntu MATE site
So what is the difference between Gnome and MATE?
MATE is a fork of Gnome 2 and retains features from Gnome 2 such as the file manager, appearance preferences, panel and indicator function(re-branded) and is not interchangeable with GNOME Classic.
The forked programs have been renamed, with most names in Spanish.
MATE DE (Desktop Environment) is a piece of software separate from Ubuntu, originally a fork of the older GNOME 2.x DE.
Ubuntu MATE, on the other hand, is (from the official page) "A community developed Ubuntu based operating system that beautifully integrates the MATE desktop."
Basically, MATE is the DE - it provides the GUI functionality.
Ubuntu MATE, on the other hand, is a derivative of Ubuntu, a sort of "child OS" based off Ubuntu, but with changes to the default software and design, most notably the use of the MATE DE instead of the default Ubuntu DE, Unity.
Source: What is Ubuntu MATE? | Ubuntu MATE