Lubuntu is Ubuntu with the LXDE desktop environment instead of GNOME/Unity.
Lubuntu also contains some different default programs, like AbiWord and Gnumeric instead of LibreOffice Writer and LibreOffice Calc, and Chromium instead of Firefox.
Whatever flavor of Ubuntu you start out with (Ubuntu, Kubuntu, Xubuntu, Lubuntu, Edubuntu, Ubuntu Studio, Mythbuntu), you can always install a different desktop environment to get the interfaces and applications associated with another one. So if you started with Ubuntu and installed lubuntu-desktop , you'd get the Lubuntu interface and applications installed, and could select Lubuntu Desktop as your session type on the login screen (while still being able to use Ubuntu's regular interface and apps).
Cinnamon and MATE are two other desktop environments, originating in Linux Mint but not specific to that OS. There's no special Ubuntu derivative based on them, but you can install them in Ubuntu. The following situations are therefore basically equivalent in terms of functionality and performance:
- Installing Cinnamon (or MATE) on Ubuntu and selecting Cinnamon (or MATE) as your session type.
- Installing Cinnamon (or MATE) on a combined Ubuntu/Lubuntu system (i.e., Ubuntu with lubuntu-desktop installed or Lubuntu with ubuntu-desktop installed) and selecting Cinnamon (or MATE) as your session type.
- Installing Cinnamon (or MATE) on Lubuntu and selecting Cinnamon (or MATE) as your session type.
Whatever interfaces you start with, Cinnamon (or MATE) will run the same way when selected (barring any bugs). It should perform the same on Lubuntu as Ubuntu--no better, no worse.
If you use different apps with it--for example, if you use Chromium on Cinnamon (or MATE) instead of Firefox--then of course those applications may perform or function differently.
Cinnamon and MATE will not perform or behave the same as each other of course; they are two separate desktop environments.