The prior icon was an Apple logo next to the word "TV," the branding that Apple uses for its Apple TV set-top boxes. The new logo resembles the Apple TV Remote, making it clearer what its function is.
Today's revamped logo matches the updated logo for the Apple TV Remote option in the Control Center on iOS devices. Apple also changed that icon from the Apple TV branding to a remote control design in the iOS 12.2 update.
iOS 12.2 brought other changes to the Remote option on the iPhone and iPad as well, introducing a design that is darker, more streamlined, and takes up the majority of the iPhone's interface. With a full Apple TV Remote available in Control Center on modern devices, there's little reason to download the standalone Apple TV Remote app.
Apple is, however, still keeping it around, and according to today's release notes, version 2.2 brings "general performance and stability improvements" along with the new app icon.
The Apple TV Remote app can be downloaded from the App Store for free. [Direct Link]