Applications don't get special permissions based on their installation directory. They inherit your permissions, and can elevate theirs (sometimes permanently AFAIK, see below) with a password prompt, but that's it.
Applications that come with an installer should stay where they are, they might break otherwise (example: FaceTime breaks if moved from
The App Store probably installs to
/Applications because it is the most well known location. It might also allow other users on your machine to use these applications. Applications from the App Store aren't actually allowed to require elevated permissions, so this is not the reason they're installed in that directory.
Applications without installer, that are installed using drag&drop, can still be placed pretty much where you want. Some of these applications fail if you use FileVault, since they must reside on the boot volume and require elevated permissions, but it's been like this for a long time and especially not related to the App Store (SuperDuper comes to mind, IIRC).
~/Applications/ is designed to hold your personal applications according to Apple (also here).
I install all applications that don't come with an installer in
~/Applications and haven't had any issues (except the aforementioned applications like SuperDuper). I think you can continue to do so yourself, but aside from file organization and not sharing these applications with other user accounts on your machine, there's not strong reason for either option.