There are two different software apps in Ubuntu 20.04, Software and Snap Store . The Snap Store and the Software app can be installed alongside each other without removing either app. If Ubuntu Software got removed in 20.04 it can be reinstalled by running
sudo apt install gnome-software. If the snap-store is not currently installed it can be installed if by running
sudo snap install snap-store.
Ubuntu Software icon in Ubuntu 22.04:
GNOME Software icon in Ubuntu 22.04:
All the apps in the Snap Store are snap packages. If you're searching for a snap package it's more convenient to search for it in the Snap Store, because sometimes the same app has two packages in Ubuntu Software, an apt package and a snap package, but the same app has only one snap package in the Snap Store. If you're searching for an apt package and don't want to use the terminal you may want to install the old Ubuntu Software app with
sudo apt install gnome-software. When you install the gnome-software package in Ubuntu 20.04 it is named Software, and the app that was named Ubuntu Software in the default installation is renamed as Snap Store. In Ubuntu 22.04 and later the Software app's name has been changed back to Ubuntu Software and clicking the orange shopping bag Ubuntu Software icon launches the Snap Store instead.
The Snap Store itself is a snap package, so it can be installed in any Linux distribution that supports snap packages without causing package management issues. snapd has been ported to a wide range of Linux distributions to enable snaps to be used across any Linux distribution, including Debian, Kali Linux, Raspbian, Fedora, CentOS, Arch Linux, Manjaro Linux, and openSUSE.