^ Agreed.
Your new computer cant connect because your server is treating each nic card as a seperate network and for sercurity reasons it defaults to not combining them.
You can bridge the two connections but as bill001g said, a switch would be a much better long term solution.
Ideally you would run a line from your router to the switch, and then plug all of the computers and server (if the server is not a dhcp server) into the switch.
Now if it is not possible to run an ethernet cable to the new pc and the server does not provide DHCP/DNS then you can just take the ethernet cord going to the server, dissconnect it and plug it into the switch, then plug a cable from switch to server, and a cable from switch to PC.