I think you're seeing your WAN IP when on your cable modem and a LAN IP when you're on your router.
A router routes traffic between two networks (in a typical setup, a WAN side and a LAN side). When you directly connect your computer into your cable modem, you are directly connecting to your provider's network, and get one IP that they have assigned.
When you connect your router to the modem, the router gets that WAN IP, and it routes traffic between the provider and the devices that are connected to the router. Those internal devices all have different local IP addresses so they can be uniquely identified and talk to eachother. The router uses a NAT (network address translation) table to send traffic to the right device.
If you are getting a different WAN IP address when you switch devices that are plugged into your modem, it means that your ISP has set your service up so that every time a device requests an IP, it will give it a new one. You can request that your service come with a static IP, so that your WAN (Internet) IP address never changes.
Ah, so this is the core concept behind "MAC spoofing", isn't it? – oscilatingcretin – 2013-10-09T20:08:46.757
1Yes, in some cases MAC spoofing will be used to obtain the same IP but in most cases it is used to overcome network security where only specific MAC has access. I would not call it concept though – TomEus – 2013-10-09T20:49:34.993