I am heavily streaming online movies from websites, and it is very handy to just find what i want using cell phone and press one button to send it to the chromecast.
I do not think it is correct to compare Fire to Chromecast, but i am comparing Fire to MiBox. Yeah, mibox cost more, but difference is not so big and it is a complete solution, with google playstore and Chromecast. The only one thing is missing in mibox is amazon prime video. But honestly, with their crappy video collection i am not using them anyway (dispite the fact that i sideloaded shield's version apk and it works ok)