None of the NOW branded boxes or NOW Stick have the Apple TV streaming service App.
What two devices are you using?
Here's a list of supported devices for Apple TV App, where Roku is not the NOW branded boxes or NOW Stick.
What is the make and model of the other smart device with the Apple TV App on it ?
I am assuming it's not another NOW TV box or NOW Stick, because they don't currently support the Apple TV App.
Just fired up my NOW black box and did a system update to double check if NOW have added the Apple TV App without my knowledge and nothing in the NOW box Apps Store for me.
@Halesintheplace is the other device a NOW box or some other streaming device you can get the Apple TV App on ?