Ubuntu software centre brought many new features like snap apps and snap store in 20.04 and those are great. But it never has an “apps and updates” page for individually update apps through software centre, like in the other app stores like google play store,windows store etc. It will be great if we got notified about application updates by ubuntu store and install updates via Ubuntu store 
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The Ubuntu Software package (which is a snap version of GNOME Software) does have this functionality. Just press the button in the window top bar that says ‘updates’.
You can press the ‘refresh’ button to look for updates.
In stock GNOME, this is the only means that is used by the system to push updates. You can choose to update individual packages or ‘all packages’, just like in your picture.
For one or the other reason, Ubuntu still uses the ‘old’ Updater programme as the main means, although I do sometimes get a notification to update via Ubuntu Software. The ‘Ubuntu Updater’ app has become redundant, I assume it is still there because it is meant to give Ubuntu users a tool that they’re familiar with.
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Personally, I would prefer that all updates are done from within Ubuntu Software, rather than Software Updates (like Fedora does). In my experience, the updates tab within Ubuntu Software seems to work less with each release. If the two were to exist, I would at least expect them both to see the same updates.
arjunvk’s mockup is a good suggestion. In addition to this, I would split system and app updates within the same window, adding an “update all” button and “update automatically” check box to each section.
Also, within the “Installled” tab, being able to see which apps are snaps, apps added by ppa, etc. This could be shown by an icon under the remove button, app size on the right hand side.
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