Mirroring Issues with Apple TV
Your question doesn't explain why you're wanting to connect the Apple TV via Ethernet, but your Apple TV and the devices you want to be able to airplay to it need to be on the same network. It seems to me that the act of plugging in the Ethernet cable is changing the network it's connected to.

If the Wi-Fi network and Ethernet cable are all connected via the same router you should be okay, but that could depend on the actual router (but most modern routers should be fine).

You need to determine if in fact they are on the same network. So, the questions are:


where does the Ethernet cable go to?
is it plugged into the same Wi-Fi router or into something else? 


Another potential issue is if you're using a VPN. For example, I've seen users with a Wi-Fi network that all devices are connected to except for an iMac and the Apple TV. Instead, the iMac is connected via Ethernet to an Ethernet Wireless adaptor (so they're connecting their Mac to the Wi-Fi network via an Ethernet cable) and then they use that connection to connect to a VPN. Once connected they share their Ethernet connection via the Mac's Wi-Fi and then connect their Apple TV to the iMac's Wi-Fi network.

While all devices in the above scenario actually connect to the internet via the same original Wi-Fi network, as far as the setup is concerned they're different connections. So for a device to airplay to an Apple TV in the above scenario they also need to connect to the Mac's Wi-Fi network, because that's the one the Apple TV is connected to.

Hopefully this info will shed some light on what's going on with your setup.

Let me know how you go. More than happy to help you troubleshoot this.