if you are connecting to a router for inet access, ipconfig will show you your private ip address. This is not the ip address that the isp gives you, which is called the global ip address. Other machines on the inet do not use your private ip address to connect to your pc. whatsmyip.com and other services show you your global ip.
when you setup a router on your home network, you creating a separate "private" network. You use the router as a gateway to the inet. Your router connects directly to the inet and is assigned the global ip from your isp. The router implements a function named Network Address Translation (NAT) to connect multiple devices on your home network to the inet.
The router is the middleman and is used to route inet traffic from the global network (Inet) to your private network.