Hi Eric,
I understand the app you added is Corporate News App. After further researching the app details, this app provides features to add different kinds of corporate news inside your SharePoint site. Therefore,I can add this app to my team site without errors, but I could not add it to my public-facing website. The App Parts I added to my public-facing website previously were some App Parts provided by default.
Given the situation, I would like to specify that the public-facing website is available to anyone on the Internet. So, everyone can visit your public-facing website without authenticated. The authenticated users have some permissions of managing the public-facing website, such as editing, adding, deleting, etc. However, your team site is a private site collection and users cannot visit it in an anonymous mode.
If this explanation did not make sense of this problem, please feel free to correct me.
Thanks,
Maggie Li