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I understand that WiFi as opposed to Ethernet connections from laptop to Airport Extreme produce a lack of internet or server connection.
But that the problem is not owing to lack of connection by way of the WiFi encryption key or 'passphrase' (also known as 'wireless password'). The WiFi connection to the AirPort Extreme is made without problems.
Three possible things:
1. the Airport Extreme is configured to block all connections but a few, by MAC address. Log into its setup and check on such WiFi security measures.
2. the Airport Extreme isn't actually being connected to ??? in which case, what you connect to by WiFi may be some other WiFi box, that is not connected to internet at all. (This is unlikely - but check that you are actually connecting to the correct SSID, and the one SSID of its kind within WiFi scanning range.)
You could make sure of this by setting a different wireless passphrase on the Airport Extreme, and then connecting to it by WiFi with that new passphrase. If there was a wrong-SSID-connecting going on, that would make sure that it could not work, and only the right WiFi box or Airport box could be connected to.
3. your Airport Extreme might perhaps be defective. Not being able to deliver internet through WiFi connection in this manner is a rarer malfunction, but could happen.
Of course, try a total reset of the Airport Extreme, by holding down its reset button for 40+ seconds continuously. Then see if WiFi Internet connection works. If it still doesn't, you have a defective Airport Extreme, it seems.