Differences between Ubuntu Software Center and Ubuntu Software & their pros and cons
What are the technical differences between "Ubuntu Software Center" and "Ubuntu Software"?
How to use them effectively or when to use what?
Is there any pros & cons for them? If yes, then what are they?
"What are the technical differences between "Ubuntu Software Center" and "Ubuntu Software"?"
Functionally, they are, more or less, the same, both pull software from your repository’s and install it.
ubuntu-software-center was developed years ago for Ubuntu and is considered slow and dated.
ubuntu-software is actually the gnome-software app and is a more modern software app.
"Is there any pros & cons for them? If yes, then what are they?"
ubuntu-software-center is slower, older and replaced by gnome-sofware even by Canonical in the main Ubuntu release, that is it for objective pros and cons as far as I know.
"How to use them effectively or when to use what?"
Simply use the one you like best, they do the same thing, like I said ubuntu-software-center is older, no longer in development but if you prefer that one, it works just fine.
Of course, the UI is different but a better UI is very subjective, I can't tell you what you would like best, Gnome is a lot more minimal and the software centre could be considered "cluttered" by some.
Only there is a difference between GNOME Software Centre (gnome-software) and Ubuntu Software Centre (software-centre)
Of which GNOME Software Centre is created by the gnome project where they added support for Snaps to GNOME Software Centre instead of Ubuntu Software Centre
Meanwhile Ubuntu Software Centre (software-centre) is created by Canonicals as a user-friendly way to discover and install apt packages (friendlier than Synaptic, which was the package manager at the time).
Where as the latest Ubuntu release is shipped with gnome-software by default.