As I recall, iBook G4's had the AirPort Extreme (802.11g) card, whereas iBook G3s had the original 802.11b AirPort card. So it's good you have a G4 because that card was capable of WPA and WPA2, whereas the older card could barely do original WPA.
Note, however, that the AirPort Extreme card in your iBook G4 was only certified compliant with the original WPA, not WPA2. Apple added WPA2 support later but never got that feature certified by the Wi-Fi Alliance, so it's possible there are incompatibilities. Then again, without knowing the make, model, hardware revision, and firmware version of the 802.11 home gateway you got with your Verizon FiOS service, I can't check whether your 802.11 gateway was certified WPA or WPA2 compliant either. For example, I know that Verizon often gives people ActionTec MI424-WR home gateways when they set you up with FiOS service, and the original revision of that product wasn't certified for WPA2 either. Only "Rev B" and later are certified.
It's interesting that you say you can type your password and it never enables the OK button. The only reason I've ever seen it do this is when your password is not the minimum length of 8 characters that the WPA and WPA2 standards require. Is it possible that your Verizon FiOS 802.11 gateway accidentally let you violate the spec and set a password that's only 7 characters or less? If so, try changing your 802.11 network's WPA2 password to be 8 characters or more and see if that fixes it.
If your 802.11 home gateway is currently configure for WPA2-only (a.k.a. AES only, AES-CCMP only), you might consider configuring it for WPA2 Mixed Mode (a.k.a. "AES and TKIP", "WPA2 + WPA", etc.), or pure WPA mode (TKIP only), to see if that old AirPort Extreme card handles it better.
Unfortunately, once doesn't exist natively in Windows and I do not know of any third party tools that do so.
While the "netsh" utility did exist in XP, Windows did not have the "wlan" functionality added to it until Windows Vista.
On Win7 or later, from the command prompt, enter "netsh wlan show interfaces" and you will get what you want. This may work on Vista as well, however I haven't been on a Vista machine in a long time and as I recall it doesn't have the more complete "wlan" functionality of later releases so I am unsure if this was included in the functionality.
Best Answer
This was a concern for a friend with similar old (but still working 10+ years later) Wi-Fi enabled equipment that doesn’t have any updates to WPA2. The old handheld device actually had better Wi-Fi range than just about any current Android phone tested, and was good for free/open Wi-Fi networks like in most stores/restaurants.
Your router might have a "WPA/WPA2 Mixed Mode" that would let you connect with WPA/TKIP or WPA2/AES. My router lists it under the Wireless security as "WPA / WPA2 Personal" and Encryption "TKIP / AES".
My friends' solution was a router that had built-in “dual SSIDs,” where you can have your regular WPA2 “N” network, and also have a Guest network that used WEP or WPA, and “B” or “G” speeds.
Or using a second wireless router hosting the WEP “G” network would work too. An excellent use of an old slow router that otherwise might get tossed away, and you could even unplug it when not using the old device, to lessen the security concerns of having a WEP network. Also, the Guest network or secondary router could be kept separate from your regular network, so even if an unwanted guest got access to it, they wouldn't be able to easily connect to your other devices.
Here’s a few links that could be helpful, centered on using DD-WRT if your router supports it (other firmware like Tomato or OpenWRT should be very similar):