@N0rbert posted a fantastic answer about the capabilities of Ubuntu MATE and its differences from Ubuntu Desktop with GNOME.
My answer has to do with the second part of your question:
I installed Ubuntu 20.04 and then ubuntu-mate-desktop, is it correct? Or there is an independent version of Ubuntu MATE as an operating system?
Yes, there is a slight difference.
Ubuntu comes in several official flavours. One of the official flavours of Ubuntu is Ubuntu MATE. These flavours are all official releases of Ubuntu that use the same repositories as Ubuntu.
The reason that Ubuntu comes in so many different flavoured releases is because changing a desktop environment on an installed system tends to be very messy. A desktop environment usually consists of dozens of packages, all with different configs that may not play nicely with a different desktop environment. Changing a DE on an installed system may or may not cause problems.
If you know that you want to use the MATE desktop environment from the beginning, then you should probably not install Ubuntu desktop and then install the MATE desktop from your package manager. Instead, you can simply install Ubuntu MATE by using the Ubuntu MATE ISO to install the operating system.