Are you both connecting with Gigabyte ethernet?
Have you swapped the ports on the router just to see if that makes any difference (doubtful but easy to check)?
What OS do you have? The memory should not make a major difference in connection speeds, I just ask because if you have only 1 GB and are using Vista or Win 7, then you may already be taxing the system. What programs do you each have running? To compare background processes as well as applications open, look on the processes tab on Task Master, or you might run the system checkup tool at PC Pitstop that lists processes running.http://www.pcpitstop.com/
If that does not help. please provide a link to the specifications for both machines.