To try to answer all your questions and for the benefit of anyone else who comes accross this thread in the future:
Q- Are these new software distribution platforms eventually going to replace the existing software centers?
A* Probably. Each distribution will, as always, do their thing. But for some the aim will be to use Universal Packages to streamline package management and reduce the work of maintaining multiple repos or package formats.
Q- What makes Snap/Flatpack so much better than current software center platforms?
A* Everything needed for the app to run is in tha single package including dependencies. This is beneficial as it can solve issues of dependency depreciation or updates breaking things. Having said that, some distros (like Ubuntu) are working Snaps into the Software Center alongside DEBs for a seamless experience.
Q- Is the idea to create a unified software distribution platform that is widely adopted across all distros?
A* Yes, but like before, each distro will do what they think is best.
Q- What’s the difference/advantage/disadvantage between installing the same software/application through Snap, Flatpack, or the distro’s own software repos?
A* Universal Packages (Snaps, Flatpaks and Appimages) have everything you need in a single bundle. There are advantages and disadvantages between the three but that is trivial in my opinion unless you care (read the linked article above for that). The overall advantage is that a single build can run on multiple distros saving developer time and distro resources. Installing from TAR, DEB or RPM etc needs the packages to be built for your specific distribution (sometimes even for each revision as well) and will need dependencies to be resolved and installed. This can be hit-and-miss sometimes.
As an example, a piece of software may require 5 DEB builds to cover the 2 recent releases of Debian and the last few releases of Ubuntu to cover the varying dependency descrepancies between them. A single Snap build will work on them all.
Another example is if you’re on Ubuntu with Gnome (using GTK for most things) and you want to install a KDE app - from the repos you will install a DEB and a whole bunch of KDE dependencies that may take over your system (once the KDE Terminal app replaced my pre-installed one) but using a Universal Package keeps the KDE cruft in check and the system untouched.