What is MATE DE?
quoting from the MATE Wikipedia article
MATE is a desktop
environment forked from the now-unmaintained code base of GNOME 2. It
is named after the South American plant Yerba mate and tea made from
the herb, mate. The use of a new name, instead of GNOME, avoids
conflicts with GNOME 3 components.
But what is a Desktop Environment?
quoting from the Desktop Environment Wikipedia article
In computing, a desktop environment (DE) is an implementation of the
desktop metaphor made of a bundle of programs running on top of a
computer operating system, which share a common graphical user
interface (GUI).
A desktop environment typically consists of icons, windows, toolbars,
folders, wallpapers and desktop widgets.
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
A community developed Ubuntu based operating system that beautifully
integrates the MATE desktop.
Ubuntu MATE is a stable, easy-to-use operating system with a
configurable desktop environment. Ideal for those who want the most
out of their desktops, laptops and netbooks and prefer a traditional
desktop metaphor. With modest hardware requirements it is suitable for
modern workstations and older hardware alike.

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.
- Caja (box) – File manager (from Nautilus)
- Pluma (quill) – Text editor (from Gedit)
- Eye of MATE – Image viewer (from Eye of GNOME)
- Atril (lectern) – Document viewer (from Evince)
- Engrampa (staple) – Archive manager (from File Roller)
- MATE Terminal – Terminal emulator (from GNOME Terminal)
- Marco (frame) – Window manager (from Metacity)
- Mozo (waiter) – Menu item editor (from Alacarte)