This isn't a "problem" in the true sense, but it is an issue that often catches people off guard. I'm guessing you're not connecting to Comcast through a router. (More on this later) The reason why you can't connect on your new computer is because Comcast's network recognizes your computer by it's MAC address.
http://en.wikipedia.org/wiki/MAC_address
Since your Macbook has a different MAC address, it is not recognized by Comcast's network, so you don't get access. If you want to deauthorize your Mac Mini and authorize the Macbook on the network, it usually requires a phone call to Comcast - the tech will ask you for the new MAC address, and they'll update their system. But you can't just swap the cable back and forth.
If, however, you want to still use both computers to connect, you will need to invest in a router so that you can share the single incoming connection amongst all your computers.