tl;dr: IMO, no

I've got the previous version of Apple TV and a Chromecast. Long story short, I use both devices exactly the same. The only differences really is how I communicate with each device - remote control for Apple TV vs phone/computer for Chromecast. The Apple TV does do streaming much smoother than casting from Chrome, and of course you can use any window, not just a website.

I think like most things though, it depends on the person. I'm a medical student, so it's not like I have free time to kill playing games on a new platform or discovering new shows.