Thanks for your suggestion. Upon further digging, I found the file location in which the binary files are stored:
However, there was only one App here (the newly created App I could open in the OP).
I then checked the Recycle Bin and found that there was an App folder in there. I restored this App Folder, found its location to be C:\QlikShare\App (the QlikShare folder I created during QlikSense Server Intallation), and this contained the binary data files for the apps that did not work.
I copied the entire contents of the App folder (and subfolders) to the new location (C:\ProgramData\Qlik\Sense\Apps) and it all works now.
Thanks for your help. If anyone could explain why the folder location changed (I didn't change it) that would be interesting to know.