Fire Stick is not compatible with Google Play Movies, but Chromecast can stream content from Amazon Instant Video. Screen mirroring is available on both devices. However, Chromecast is much better than Fire TV. Both Chromecast and Fire Stick are compatible with the game, but Fire TV has a larger compatible selection.
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You will always need a mobile phone to control Chromecast. Amazon Firestick allows you to connect Bluetooth headphones were using them in Chromecast is a tedious process.
Firestick supports all streaming services like Chromecast.