Guess your PiAware site is on the same network as the computer you are trying from. If it is not things get more difficult.
Your Pi does show to be feeding data to FlightAware so it is working.
Go to your MyADSB page and confirm that your router has not reassigned the pi network address. This is found on the site information block and is probably the last line there. Incorrect IP would also account for your problems with other applications. Routers typically use DHCP for assigning addresses and tend to change the IP of any device according to whatever rules that are in the router. I keep my pi set to static IP and also assign that IP in the router so it does not duplicate the address on other devices but that requires more that I can post here.
When you switched to wired connection, if my memory is correct, that would be another IP address. The wireless IP and wired will not be the same.
There are several other ways to confirm sites IP address even if its running without local monitor and keyboard which I’m not touching on here.
Without trying here seems like my memory is that when you hook up wired the IP is not the same as the wireless on your Pi.