Ubuntu vs Ubuntu Mate

I lately installed Ubuntu 20.04, and according to an answer about gnome Flashback I installed ubuntu mate.

I wonder what benefits does it give, I work with it a couple of weeks and still don't see a lot of change (except of visual interface).

Can somebody please clarify the difference?

BTW, 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?

The basics: MATE is desktop environment, Ubuntu MATE is an official Ubuntu flavor.
One can do fresh install of Ubuntu MATE using official ISO or by installing ubuntu-mate-desktop^ (task) package on top of existing system.

Then about the differences and benefits.
The complete list depends on your workflow and needs.
On my daily work in Ubuntu MATE I'm happy with following functionality:

• Caja file manager for files and desktop:

  • have any objects inside my ~/Desktop folder, controlled by Caja (MATE file-manager, see screenshot here);
  • have fully functional drag-and-drop between any folders including ~/Desktop and MATE Panel;
  • have Dropbox, Git and Mercurial repositories integrated into Caja (see screenshot here);
  • have fully integrated Engrampa archiver to create password-protected archives;
  • have an ability to launch applications by visiting /usr/share/applications by Caja;
  • have a side pane opened by F3 in the Caja with various modes;
  • have infinite possibilities to expand Caja by scripts and actions;
  • have 400% large thumbnails.

• MATE desktop overall:

  • have an ability to use Applications, Places, System menus on the top MATE Panel;
  • have an ability to set Ctrl+Shift keyboard shortcut to switch keyboard layouts with my PPA to prevent interference;
  • have keyboard access to the window frame mnemonics by Alt+Space;
  • have an ability to indicate current keyboard layout using country flag;
  • have an ability to set up Compiz with all its plugins like Annotate;
  • have customizable MATE Panel without plugins and extensions;
  • have an ability to install and use tiling utility named X-tile to effectively organize screen space;
  • have an ability to leave a message from screensaver of locked system for a user;
  • have well-integrated e-mail checker into Indicator Messages by Popper (on 18.04 LTS), MailNag or Ayatana WebMail (on 21.04 and newer);
  • have an ability to install some old but cool ScreenLets and custom widgets to be shown on desktop;
  • have an ability to list calendar events from Evolution in preinstalled Clock applet;
  • have an ability to check time and weather in different timezones using preinstalled Clock or indicator-datetime applet;
  • have media players like VLC or Guayadeque Music Player integrated into indicator-sound as shown on image.

@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.