Generally the IP address of your network is what is visible to websites. Go to www.whatismyip.com and you will see what is visible. Do cmd - ipconfig on your PC to see what your computer's IP is.
The difference is handled by NAT - network address translation. If you google the term you'll get more info.
Changing the IP of your internet connection can sometimes be forced by resetting the router. For a corporate lan, you probably have a fixed IP though. Some home accounts such as cable, I've seen them tied to the MAC address of your cable modem, so I wouldn't mess with forcing a MAC address change.