Can't use Wi-Fi and LAN at the same time
You can change the priority for Wi-Fi over LAN connection. By default machines have LAN connection at top priority. You choose Wi-Fi at top and try again.

Step to change the priority of connection;


Go to System Preference -> Network, in network section at left-below side you have three icons +,-,(icon for setting). 
Click on setting icon and you will get multiple option.
Select "Set Service Order..."
Then you will be prompt to a new window with all network option.
Select Wi-Fi and drag it to top.


It should help you to get your internet connection back. Please let me know if it works.