What is the difference, and can I install the Ubuntu 16.04 LTS on a Pi 3 Model B?
The difference between Ubuntu and Ubuntu MATE is explained on Ask Ubuntu:
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.
The Ubuntu wiki also has clear, easily accessible instructions on installing Ubuntu on a Pi 3:
Raspberry Pi 2: ubuntu-18.04-preinstalled-server-armhf+raspi2.img.xz (4G image, 296MB compressed)
Raspberry Pi 3: The official Raspberry Pi 2 images can be used with a Pi 3B/3B+ after minor changes (see instructions below).
Alternatively, you can use the unofficial image which supports the Pi 3 natively; but 18.04 and 16.04 LTS are available.