the modem will need resetting, as it only talks to 1 mac at a time, so its basically not listening to anything else other than your PC.
Reset the modem, after you've plugged the cat5 cable into the ps2, then try setting up online play...
You'll have to set the PS2 up to use DHCP, as the details you have gotten from ipconfig /all on your PC will only be relavent to your PC, as the MAC address for your ps2 is different, therefore your IP will be different for the PS2 also, as the DHCP lease will be for that MAC.