What are the differences between Ubuntu MATE and MATE? Is the DE MATE different from the DE Ubuntu MATE? Like, does MATE also stand alone from Ubuntu? It kinda seems like it's bolted on to several Linux systems. What is the difference between Ubuntu MATE and Ubuntu with the MATE desktop added?
Ubuntu is the collection of packages/programs, while MATE is the desktop environment (one of many options). MATE has been around since 2011. It is a continuation of GNOME 2 (2002-2010), which was the default DE for MANY Linux distributions.
MATE can be used with any package base, such as Debian, Fedora, Linux Mint and Arch.
The MATE DE is developed by the team that also develops Ubuntu MATE. While you can use MATE with other distributions, Ubuntu MATE gives you the "first class" experience with options not present elsewhere. This is similar to the experience of using Cinnamon DE with Linux Mint, as opposed to using it with Debian.