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If you were to check your routers external IP you would find it to be the same as the ip if you connected it to your computer. Taht is the IP that the World sees. When you connect your Router to your computer you are essentially setting up a Internal network and therefore it would be a different IP. The IP you see when you plug the network card into the Modem, is the IP that your ISP is assigning you. The IP you see when you connect to your router is the IP your Router is assigning you, and your ISP is assigning your router your old IP.
Also it could be much eaiser then this. Perhaps you are on a dynamic IP service for your ISP, and in witch case your IP would change if you changed the MAC Address (one would be your router the other would be your Network card)
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