? what? I'm not pointing anything on you. I just didn't understand the question or the answers given to you. To me it was like everybody understood the question except me and I felt awkward like if someone were hitting on my mom in front of me and I thought it was maybe because English is not my LANGUAGE OR that all the others haven't understood the question neither.
As you can have more than 1 DE in your system it may be that what you was asking. Or maybe you don't like none of those DE and you was asking why people use those two...
Now I can tell you the following:
Ubuntu MATE is UBUNTU MATE group and MATE is another group. I know you are saying what? So I'll show the path to go thru.
In Linux World (not Linux) there are thousands of groups because Linux is community collaboration.
Linux is an OS that is developed by developers all around the world. Lots of them are actually not doing it by their own but working for a company that makes a team work on Linux. There a lot of companies that do that. Also there are a lot of developers that donate their free time to work on Linux.
Linux is formed of lots of pieces. The Linux kernel is the first one but it has branches, why? Because you don't need lot of stuff to put in a router (oh, yes, most houses routers are working in some version of Linux. Moreover, Android is Linux, so most mobile phones use Linux).
Then you have CLI tools. Then you have X (and others). Above X you have a Windows Manager or a DE with a Windows Manager (MATE, KDE). Also, before seeing the DE or WM you have a greeter, e.g. LightDM or GDM, the one that ask you for a password.
Then you can use a lot of different software e.g. Pluma, Gedit as text editors or LibreOffice, Apache OpenOffice or WPS.
So, having all hat into account, there's is some|thin/one like Canonical that chooses a Linux kernel branch + a Greeter + a DE + other Software and distributes it (also they can make changes to the software). That is called a LINUX DISTRIBUTION. The one from Canonical is called Ubuntu and it has different versions, 16.04 the last LTS release. Then, as of Canonical, offers can be that the base is almost the same but with another DE and a little pieces move back and forth. The latter are called FLAVOURS and Ubuntu MATE is a Flavour of Ubuntu (and thus from Canonical).
In the end, I can tell you the following: Ubuntu MATE is a "group" part of Canonical but MATE is not a group part of Canonical neither Ubuntu MATE.
TO complete my answer, there are other distributions: Mint, OpenSuse, CentOS, Arch and a lot more.