Airport Extreme only works behind a router

Question

Yesterday my cable modem suffered some kind of malady, rendering it inoperable. Don't know what happened, but it just stopped working. Time Warner replaced it, but now I've got the following problem.
The original cable modem went to an SMC Barricade router, and one of the LAN ports was connected to the WAN port on my Airport Extreme. Other ports on the router were used for connecting an old G4 (no wifi card) and a printer (Laserwriter 320 with Farallon iPrint interface). Our laptops connected to the Airport Extreme and the G4 accessed the internet through the router.
With the new cable modem, if I connect the router to it, nothing can access the internet. Putting the G4 on the cable modem by ethernet works, so the cable modem is working. I think whatever took out the cable modem, took out the SMC router, too. I can't even get the G4 to print through the router, so I think it's toast.
When I hook up the Airport Extreme to the cable modem, my laptops can see the Airport network fine, and under network status it says they are connected to the Airport network and are on the Internet, but no web page ever loads and eventually I get a statement in my browser that I'm not connected to the internet and to use Network Diagnostics. Network Diagnostics never finds and error and says to contact my ISP. HOWEVER, if I connect the G4's ethernet cable to the LAN port on the Airport Extreme, the G4 can access the internet, no problem. It's like the Airport Extreme will allow an ethernet LAN connection to the internet, but not a wireless connection. And I've got great signal strength and by all appearances everything should be working wirelessly, but they're not.
Then I remembered I had my sister-in-laws NetGear N wifi router in storage (I wish I could keep it, but I'm holding her stuff while she's moving and she'll want it back). I hooked it up to the cable modem and we could access wireless internet, and the G4 and printer could as well through the ethernet ports on the router. So it's not an ISP issue, it's an Airport issue.
On a lark, I hooked up the Airport Extreme to one of the LAN ports on the Netgear router. Now my laptops can access the internet through the Airport Extreme. ??????
Why is it that I can wireless connect to the internet when the Airport Extreme is going through the NetGear N router, but not when it is connected to the cable modem directly? With the original cable modem it was going through a router, too. Why won't it work when wired directly to the cable modem?
Update -- I did see that connection sharing was set to off (bridge mode). However, when I change it to "share a public IP address" and update it, I get an error that the DHCP address conflicts with the WAN IP address and to change it. The WAN IP of the Airport Extrme is 192.168.2.2. I've changed the beginning and end DHCP addresses to numbers much larger, like 192.168.2.100 - 192.168.2.200, but it still tells me that's a conflict. I don't see how that range could be a problem, so what am I missing?

Answer 1

I performed a reset on it from within the Airport Utility, but got an error after updating saying it couldn't connect to the AEBS. No signal from the base station. Hmmmm.
Then I pushed the reset button on the AEBS itself. I still couldn't see it wirelessly. I hooked up an ethernet cable to the LAN port on the AEBS, then I could access it.
I put in all the old values and I could now get a signal from the AEBS, so I disconnected the ethernet cable. But for whatever reason I couldn't access any web pages wirelessly, although the Network pref pane said I was connected by Airport.
I power cycled the cable modem. No change.
I checked every setting I could in the Airport Utility, nothing seemed amiss, but it wasn't getting an IP address. Embroiled in frustration, I restarted the base station.
And Voila! I can now access the internet wirelessly.
So, I think some serious ghost got in the machine, but a couple of resets, and a restart, and it seems to be working.
Next, let me hook up the ethernet printer and see if it "sees" it over the network. Ah, yes it does.
And I swapped an old airport card out of a laptop and put it in the G4, so it's wireless now, too.
Problem seems to be solved.
Thank you very much for your assistance.