Can connect to wireless and wired network, but not the internet

I'm not an expert but:

Does it work when you connect the Wi-Fi from another computer (i.e. a laptop)?

And what about restarting the router?

You've posted here, so I'll assume you can still connect a different
machine / device to your network and get the Internet at the same
time you can't with the Trisquel box.

My first guess is you've done exactly as you've implied in the thread
topic and had both wireless and wired connections active together at
some point. Even if there's only one at present use the Network
Manager panel / taskbar applet to disconnect all current network
connections then reconnect just one when the disconnected
notifications have all finished.

If you've not been caught out by this 'power user' feature and your
Internet still doesn't work then we need to find where exactly the
problem is. You'll need your router's IP address for this which,
unless someone's changed the default, will be printed on one of the
labels somewhere on the router. IP addresses are easy to spot
because they're written as four decimal numbers between 0 and 255
separated by dots. Most home routers will start 192.168, so look for
something like 192.168.1.1 . Now open a Terminal window (it's in the
Accessories menu) and type the following three commands on one line
each, following each by pressing the enter key.

ping -c 3 "your router's ip address"
ping -c 3 208.118.235.148
ping -c 3 gnu.org

Make the obvious sustitution for "your router's ip address" of
course. Then copy and paste the commands and their output into a
text file on a USB stick (there's a text editor in the Accessories
menu), open it on another machine and copy and paste it again, this
time into this thread.