The most obvious advantage is the app doesn't have to learn the users new password when they change passwords.
But you're right, the whole dance of having the app never `touch` the user's password is bullshit, a pretense cooked up by bureaucracy. You could get all the meaningful functionality of Oauth by letting apps request a single "permission key" when they login - with the user's password. On the other hand, if a site want real, meaningful security, it could give each user a seperate "app password" that they authorized apps with. That would provide real protection - but since it requires one teentsy extra step for the end user, it will never, ever fly.