Do I need the Apple TV app for my Apple TV subscription?

You can use the website to watch Apple TV content, but most prefer to use an app. The choice is yours.

I honestly don't know the difference or if there is a benefit to the app. I believe the app first appeared built into Mac OS with the Catalina update in 2019, so if you do not have it, then I suspect you are using a Mac OS version older than Catalina. It has just been in the last couple of years that the app was made available on Windows PC and Android devices, so previously those users had no choice other than to view Apple TV content in a web browser.