If you've ever travelled internationally and used Uber, you probably didn't see much of a different experience. This is deliberate. There's no ‘Uber: North America' or ‘Uber:China'. It's just the one app, worldwide.
You may also be mistaken in thinking Uber's app is only a handful of screens. However, according to someone from their engineering team, it's anything but -
I worked on payments, and this is what screens and components are in the Uber app:
Despite having to overcome some very big engineering hurdles, Uber insists on using only the one app per platform (Android / iOS). Considering the challenges that come with an application that has so many different rules, requirements, contextual differences, languages and so forth, you'd be right in thinking it'd be easier to instead have different versions of the app depending on its location.
So why does Uber attempt to jam as much into one single app, when it could have multiple apps across different regions?
The answer: Customers only ever need one ride-sharing app - so why give your competitors an opportunity to be that one app?
Consider this scenario: You've landed in a foreign country, you've grabbed your bags, you're tired, jetlagged, desperately need a shower and after a lengthy period in customs, you're ready to head to your hotel. But how to get there?
You have a decision. You can either use the app that lives on your phone (Uber), - or you can sign up to a brand new, untested, service. All you'd have to do is find it, download it (via airport wi-fi), sign-up, submit your information, submit your payment info (do they take Amex?), and then learn how to use the app.
The choice to the user is mindlessly easy: use what you've already got.
This is why Uber insists on using the one app. They don't want competitors to get a foothold in. Remember - the competition might be cheaper, have better features, work faster, be more ethical etc. Uber most likely can't compete pound for pound against its rivals. They know that it's a race to what lives on the phone and there can be only one; the first ride-sharing app you install will be (most likely) the one you use forever.
Uber doesn't want to open that the door for competing apps to be accessible. Once that happens, Uber is then in competition for every subsequent ride.