Hi aodhgan,
I have just tried a few things, and it looks like the wired ethernet isn't brought up during boot, when you run the Galileo board from the built-in flash (I have tested with firmware version 1.0.2).
If I boot the board, and upload the WebClient sketch, I get the following in the terminal window in the Arduino Environment (In theory the window should be opened very quickly, since the code will only try once, but it seems like the output is being cached/buffered so the timing isn't that important. I'm testing this on Mac OS/X):
connecting... connection failed disconnecting.
If I login via the serial console (RS232 jack) and bring up eth0 (using command ifup eth0) and upload the sketch aging, it works and I get the output from the Google search including some headers.
When doing the same with the SD card image and booting from that, it seems that eth0 is brought up during boot and when uploading the WebClient sketch, it works without doing anything via the serial console (RS232 jack).
I don't know if it's intentionally on the SPI flash image, or I'm doing something weird, but if you add in the following in the beginning of the setup() routine in the sketch to bring up the wired ethernet connection, it works for me without an SD card and without the need to login via RS232 serial console:
system("ifup eth0"); delay(5000);
The delay() is just to give it a little time for dhcp before continuing.
I hope it helps!
/Thomas