The web version of the app is hosted at ui.opensprinkler.com
When you access the controller directly in a web browser, it returns a homepage that contains re-direction link to the above site, so the browser can load the web UI from there.
The mobile app is the same Javascript code that is wrapped into a native app. The appearance is the same.
The UI looks different with different firmware because each firmware supports different sets of features and the UI detects your firmware version so that it enables or disables certain UI elements accordingly. For example, OpenThingsToken (OTC) is only supported from firmware 2.2.0, so it doesn’t make sense to show this option for firmwares earlier than 2.2.0.
The firmware itself provides HTTP API to interact with the app / web UI. You can go to support.opensprinkler.com and check the OpenSprinkler HTTP API to check the details.