The AirPort Extreme has two passwords.....one is the Base Station or Device password.....and the other is the Wireless Network password. If you have added a hard drive to the AirPort Extreme, then the drive may have a disk password as well.
If you are opening AirPort Utility on your Mac, the application normally is looking for the Base Station or Device password....not the wireless password, unless you specified during the original set up that you wanted to use the same password for both the Base Station and Wireless.
If you use the iPhone or iPad that you used to originally set up the AirPort Extreme, then it has stored the passwords that you need and enters them automatically. That's why you are not asked to enter a password when you use the iPhone or iPad.
If you have AirPort Utility installed on the iPhone or iPad, you can "see" or "retrieve" your passwords as follows:
Open AirPort Utility on the iOS device
Tap on the picture of the AirPort Extreme
Tap Edit at the upper right corner of the screen
Tap Advanced
Scroll down and tap on Show Passwords, and they will be revealed
Otherwise, you can perform a Soft Reset on the AirPort Extreme to reset all the passwords. It might be a good idea to write those down. For more info, see this Apple support document and look for the info on a Soft Reset.
Resetting an AirPort base station FAQ - Apple Support
If all else fails, then you will need to perform a Factory Default Reset on the AirPort Extreme and then set it up again.