I think the biggest difference is in how non-technical users perceive the safety and quality of the apps. The internet is a scary place with viruses and scams and loads of crappy stuff. An official app directory by Google or Apple is safer, and seems even more safe, as users perceive the content as curated and approved.

Developers, on the other hand, can charge money for the premium of having gone through the curation process. Users are more willing to pay a few bucks on an app store than for the same website when they entered the URL into the browser themselves.