Chromecast all but requires another smart device running (continuously) to control it. You can't control it directly.
Google TV, on the other hand, AKA "Android TV Sticks", are a full-on Android device, just like your phone or tablet, but without the screen. You control it with something like a mouse/keyboard.
You can turn off your phone/laptop while using a Google TV. You can browse the Internet on your Google TV, without using anything else to help. You can plug in a keyboard/mouse and use it like a computer! You buy apps on it from Google play, just like any other android device, and it's very compatible!
I just loved my first TV stick that I bought on Amazon (MK808b) for $35! I just bought an MK809 when my MK808b finally died after 3 or 4 years of daily use, and it has (so far) been a nice upgrade. Faster processor, better wifi reception, more memory/storage. Still runs just fine off the power from the USB port on the side of my TV...
PS: To control one of these, you want a "flying mouse remote". It's a keyboard that "mouses" by waving it in the air.