It is unusual to be able to swap the connection straight over from one computer (actually one "network adapter") to another without having to power off the Cable Modem and leave it without power for a couple of minutes.
This is so that it "releases" the association between the existing connection (recognised by its MAC address) and the IP address that it has allocated to it. When you power the modem on again, it finds a new connection and allocates a new IP address to it; you should then have internet access.
At the moment, your network adapter is not getting the new IP address it needs. By default, Windows will allocate a "169.254" address which will give the Limited or No Connectivity indication and you won't have internet access.