If I understand you...a cable goes from the cable modem to the Airport jack, right? And that works, but if you disconnect the cable from the AirPort and plug a computer into it, so that the cable goes from the cable modem to the computer, it does not, right?
If so, that is normal and expected.
Some cable companies use "MAC address tracking." A MAC address, which stands for Media Access Control address, is like a serial number. Every single Ethernet device--computers, AirPort hubs, routers, switches, you name it--has a different and unique MAC address.
With some cable companies, the first time you plug a gadget into the cable modem, the cable modem makes a record fo the gadget's MAC address. From that instant onward, that cable modem will never again work with any other computer or device, unless the cable company resets it.
If your cable company does MAC address tracking, that means your cable modem has locked on to the MAC address of your AirPort hub, and from now on it will never work with any other hub or computer or router or switch or anything unless the cable company resets it.