What is the difference between chromebox and "android tv"? What is the difference between Chromebox and "Android TV"?   I was thinking of buying either a Dell Chromebox or a Lenovo Touchscreen Chromebook to go with my Chromcast on my living room TV but I don't know if Chromebox is designed to work like the Chromebook.   Also, would a Chromebox be a good replacement for my Chromecast?
A chromebook (or chromebox) runs Chrome OS. This is basically a very lightweight version of Linux that runs a Chrome Browser. You do not have the ability to install "native apps", with the exception of recently, it was announced, you can now install some Android apps.

The difference between Chromebook and Chromebox is a Chromebook is a laptop, which has a screen, keyboard, mouse/trackpad. While a Chromebox is the desktop version. It does not include a monitor, nor a mouse or keyboard.

Android TV runs Android. It is designed to be connected to a large display, like a TV. Its UX is designed for the 10 foot experience. It runs Android applications, but may not run all the same apps that you can run on a tablet or phone, because of the different user experience. Apps need to be tailored to use a remote or a gamepad. 

Android TV would be a better "replacement" for a chromecast. As they usually offer the functionality that a chromecast does, and more. 

A chromebox is not necessarily designed for that 10 foot experience, and may not offer you the apps for streaming content like Android TV would.  While you could probably log in to the service websites and stream from there. But you would need to have a mouse and keyboard connected.