Verify that your Apple TV (3rd generation) has been updated to the latest version system software: Apple TV Software 7.8. Update your Apple TV - Apple Support
The TV app on Apple TV (3rd generation) should be similar to the TV app on iPhone.
There are a few differences:
- The TV app on iPhone has the tabs at the bottom of the screen, the TV app on the Apple TV box has the navigation tabs along the top of the screen.
- The TV app on iPhone allows for content from connected apps.
- The TV app on iPhone allows for Home Sharing content.
- The TV app on iPhone will be updated as needed/deemed beneficial. Apple TV (3rd generation) is EOL.
Both offer access to Apple TV+.
Both offer access to the iTunes Store for movies and TV shows for buy or rent. Both offer access to your Library for the iTunes Store.
Both offer access to premium Apple TV app Channels.