It depends on how the automatic updates are handled.

App updates can be cryptographically signed with a key that’s kept offline and only held by a few people. Your trust in the app can be equal to your trust in those people, multiplied by your trust in the technology that keeps those offline keys safe from compromise.

Your trust in a web app will be equal to your trust in the people running the server multiplied by your trust that the server isn’t compromised. Since the server is necessarily always online, that trust will be much lower.