As Long as you have the Ethernet connected up properly, it should work exactly the same as the WiFi: none of the software needs to change, the only difference is the IP addresses you will use, as the WiFi and Ethernet interfaces each have their own address.
How do you have it physically hooked up? Are the PC and the Yun both connected to a regular network? Or do you just have a direct connection between the two with no router? If the latter, you probably don't have a DHCP server to hand out addresses. If this is the case, you will either have to assign compatible but different static addresses to the PC's and Yun's Ethernet ports, or you will have to set up a DHCP server on one or the other.
By default, the Ethernet port is set up as a client to get its address from a DHCP server, and if there isn't one on the wire, it will never get an address. When using WiFi, if the Yun is the hotspot, it is also a WiFi DHCP server, and if it connects to a hotspot, the hotspot provides access to the DHCP server.