You might have already read lots of reviews about Amazon Fire TV Stick and Google Chromecast  so here is my bit of it, I Currently use both of them using Google Chromecase since 3-4 Years and Amazon Fire TV Stick  since a year now so these are based on my usage

Product	Amazon Fire TV Stick	Google Chromecast
Interface	HDMI & Micro USB	HDMI & Micro USB
Remote	Yes	No
Android App	Yes	Yes
Power Adaptor	Included	Not Included
Proxy Support	Yes	No
Cast Mobile Screen	Yes	Yes
Mobile Required	No	Yes
Amazon Prime Support	Yes	No
Youtube	via Browser	via Mobile Cast
Ability to Install Apps	Yes	No
Resolution	720p and 1080p up to 60fps	Up to 1080p 60fps
Network	Wireless + Ethernet with Adaptor Dual-band, dual-antenna Wi-Fi supports 2x2 MIMO 802.11a/b/g/n/ac	Wireless 802.11a/b/g/n/ac (2.4GHz/5GHz)
Storage	8 Gb	NA
RAM	1 Gb	NA

Verdict : After using both the devices for more than a year i strongly recommend Amazon Fire TV Stick over Google Chromecast as it comes with Remote (Voice Search) and ability to install apps you do not need Mobile phone each time to use Amazon Fire TV Stick,