Mint was a reaction to Gnome 3, the unique workflow upset a lot of people and the people behind Mint decided to build Cinnamon desktop (its Gnome 3 made to look/work like Gnome 2). They needed a distribution to build/test their work and so based a distribution off of Ubuntu and called it Mint.

As a bit of explanation, there are only a few projects which attempt to build an entire linux distribution from scratch. This involves finding code from thousands of sources, work out packaging, etc.. We call these 'base' distributions, Debian is the base distribution for Ubuntu, Ubuntu is the base distribution for Mint.

Ubuntu tends to be slightly ahead of Debian in the software versions it uses and automatically enables the 'non-free' repositories. Ubuntu tends to push some Canonical specific things like Snaps (which everyone hates)

I believe Mate rolls the Canonical specific things out of Ubuntu and you get the latest version of Cinnamon.

Its all a bit...