Chromecast: Chromecast came from Google which is based on Goolge Cast technology. Chromecast devices also play the audio and video on an external display, but it works via Wi-Fi from the internet or a local network.
So, due to the different technology with Miracase, Chromecast needs a router to work collaborately, actually Chromecast needs an existed network, otherwise, Chromecast can not work.
Miracast: Miracast is stlll a kind of Wi-Fi, but uses Wi-Fi Direct connection, with W-Fi Display technology, the Miracast defines a protocol to enable a device to wirelessly connect to an external monitor, mostly is a TV.
Because of Wi-Fi Direct, Miracast device, such as Microsoft Wireless Display Adapter, allows a smart cell phone with Miracast feature supported to transfer video and audio signals to a TV or a monitor directly, without other network control device help such as a router.
Click here to buy a Wiress Display Adapter from Microsoft Store. Note: This Adapter is NOT only support Windows phonts (actually there are few Windows phones support Miracast), Andorid 4.2 above systems support Miracast also, so Andriod phones can use this Adapter too.