are u using a crossover cable to the second computer. a crossover switches the send and receive wires on one end.
You also need ICS installed which the wizard should have done.
A router would enable you to have internet access to the other computer without the other computer being on. Plus if the main computer should lock up or need to reboot you will lose the access on the second computer.
A router will also enable future expansion or maybe lan parties if you want to.