The RouteYou PWA is a Progressive Web App.
That means it's a website that's also designed to function like a mobile app. You can also add a PWA shortcut to your device's homescreen just as you do with other apps.
The RouteYou PWA offers all same functionality available on the website.
Read more about the RouteYou PWA here.
The RouteYou App is a native app.
Native apps are apps developed for a specific platform, such as Android or iOS. You must download a native app from an app store and install it on your device before you can use it. Native apps often offer a better user experience because they are developed specifically for the device in question.
The RouteYou app has streamlined user functionality, vs the PWA, to focus on an optimizing the user experience.
Read more about the RouteYou app here.