You cannot simply move the cable over and expect it to work. When you first connected your PC and turned it on, your PC (network card) "announced" to the world (your cable modem, or router) that it was there and asked for an IP. Your cable modem then noted that MAC address of your PC's network interface card (NIC) and assigned an IP to that MAC address. When you swap the card over, the laptop's network interface has a different MAC address so the cable modem says "no way, I'm not talking to you!"

To get your laptop to work, you need to ensure it is set up to get an IP dynamically (it probably already is). Then power it down. Unplug the power cable from the cable modem. Then unplug the network cable from your PC and connect it to the laptop. After the cable modem has been unplugged for 60 seconds or so, plug it back in, wait for the lights to stabilize then power up the laptop. It should then get assigned an IP.

Note you will have to do the reverse when going back to the PC.

To get around all this back and forth stuff, and to add a great new layer of security, you should get a router. With this device, the router assumes the IP assigned by the cable modem/ISP and then it assigns new and unique IPs to each of the devices attached to it, such as your PC and laptop. Because your PC and laptop have new IPs, but the world only sees the IP assigned by your ISP, your true IPs are hidden - making it more difficult for badguys to hack your systems. These are very easy to setup and well worth it the money. Any router will do just as long as it support NAT (network address translation). They typically come with 4 or 8 ports to add computers to your network - your network being everything on your side of the router.