Again, without your exact equipment, it's hard to tell you EXACTLY what you need to do... but, it sounds to me like your DHCP or PPPOE session is not expiring, and so you're getting the same IP address.
If your ISP is giving you a dynamic IP address, this is assigned to the MAC address it encounters in your router when the modem is powered on. If you simply power off the modem, your session with your ISP is NOT CLOSED. This means the IP address you had when you turned off your modem stays locked to that modem (even if you change the MAC) for a pre-determined amount of time. This is set by your ISP, so it may be a few minutes or 24 hours.
I was having this exact same problem, until I did some digging around in my MODEM configuration pages. Here I was able to find "current connection status" or something to that effect (may be different on yours), which was listed as "connected" and then I clicked on "release" (may be disconnect or something along those lines on yours)
THEN change the MAC on your router
THEN reboot your router
THEN reboot your modem
Hope that helps!
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