Mac mini will connect to AirPort Extreme, but not internet

Are you sure that iPhone and iPad can access Internet via Extreme if you have amber indicator next to Internet on Airport Utility?

If you open Airport Utility > Select you Airport express device > Edit > Internet

Do you see a correct IPv4 address that should be provided by your Internet provider? Try to click on Renew DHCP lease.

Additionally, is there are any DNS server numbers/ IP addresses.

Because Ethernet has higher priority on your Mac Settings > Network

and since you had the cable plugged in, the Mac was trying to use an Ethernet instead of WiFi.

You may put the cable back, then rearrange connections which will change its priorities by having WiFi on the very top.

you may connect an Ethernet cable to your Extreme (it is what I have) and use faster Ethernet connection.However, if I disable or disconnect Ethernet - WiFi connection automatically becomes active. You may do it vice versa... Use WiFi (having it on the top of network device priorities) and if for whatever reason you will lose it - Ethernet connection should automatically serve you instead (if this switch is not automatic - you may need to turn wifi off for Ethernet to take care of your connections)

One of the problems might be wrong Ethernet IP configuration.

Check your working WiFi IP configuration

let say it is:

IP address 192.168.1.10

Subnet Mask: 255.255.255.0

Router: 192.168.1.1

DNS server: 192.168.1.1

then set the Ethernet's configuration to the same with exception of an IP address. The last digit of Ethernet connection IP address should be different from WiFi's and Routers... and unique on the network. Let's say 192.168.1.11

As well you may check that Configure IPv4 is set to: Using DHCP if you use it for WiFi, this should provide Ethernet with the correct IP address automatically.

make Ethernet

IPv4 Address: 10.0.1.5

Subnet Mask: 255.255.255.0

Router: 10.0.1.1

DNS Server: 10.0.1.1

Assuming that there is no other device on your network using 10.0.1.5 IP address.

If 10.0.1.5 is in use replace the last digit with anything between 6 and 254

However, the easier way would be to configure Extreme to provide IP addresses via DHCP. DHCP would provide iP addresses automatically from your defined scope and would take care of all the duplicates and other devices that may connect to your network.

have you configured your Airport extreme to provide addresses via DHCP?

If your WiFi settings are:

IPv4 Address: 10.0.1.4

Subnet Mask: 255.255.255.0

Router: 10.0.1.1

DNS Server: 10.0.1.1

and WiFi is working with no problems

then having the same settings, just different IP address' last digit should work for Ethernet...

Have you connected the Ethernet cable to the computer as well as the Airport Extreme ?

Alternatively, you may change DNS server address to 8.8.8.8