HD Audio or AC97 connector - Which to use when, and what's the difference?
The correct one to use depends on your case. Those cables are typically used to connect front panel headphone/microphone ports. While the connectors may look identical, the pinouts are different and not really interchangeable. The matching connector on your case should be labelled accordingly, I know mine is labelled HD Audio.
HD Audio is intended to replace AC'97. HD Audio is generally more capable. The linked articles have more detail on that. For average home use, there shouldn't really be a noticeable difference.
The rest of this answer will be referencing the Intel Front Panel I/O Connectivity Design Guide, page 19. They also have schematics on pages 20 and 25 if you care. I will be adding emphasis.
Front panel audio design in conjunction with motherboard audio header design is dependant upon the type of audio CODEC being used on the motherboard. In the past, AC’97 Integrated Audio CODECs were prevalent. With the introduction of Intel High Definition Audio, many new motherboard designs are switching over to High Definition (HD) audio CODECs. Designers should note that AC’97 and Intel High Definition Audio front panel motherboards and I/O cards implementations are different and may not be directly compatible or interchangeable
Some sources indicate that it is/may be necessary to switch audio output modes in the BIOS configuration or with a jumper. It is recommended that you look at your motherboard manual. My own motherboard (MSI P67A-C45) manual just says it's compliant with the Intel Front Panel I/O Connectivity Design Guide, not all that much information. Dell manuals describe it as front audio (F_AUDIO) and don't indicate any such jumper, and only describe the BIOS option Onboard Audio Controller as Enabled or Disabled (Enabled by default).
Passive AC’97 analog front panel dongles (ones which leave the 5V Analog pin-7 line unconnected on the dongle) may be used with the Intel® HD Audio analog front panel header. But note that the front panel jack detection and re-tasking functionality will be lost as the AC’97 jacks cannot support connection to the SENSE line. In addition, software must be aware that an AC’97 dongle is being used with an Intel® HD Audio analog header since the software might need to dedicate codec ports that are connected to the header to meet the product’s intended functionality.
"AC97" and "HD Audio" reffer to Intel standards for onboard audio. A minor part of those standards is a front-panel audio connector.
However the connectors are only semi-compatible. In particular the handling of plugging in of front panel devices is different.
On AC97 audio to the rear audio output was looped by the front audio output and plugging in a device to the front panel audio output would electrically disconnect the rear audio output from the audio source. If you wanted to use an AC97 motherboard without front panel audio it was nessacery to put jumpers on the audio header to make the rear audio output work.
On HDA audio to the front and rear outputs comes from seperate outputs on the audio codec. The pins that were previously used to loop back audio from the front are repurposed as connector-detect pins. It is up to software whether to disable the rear panel audio output when a connector is plugged into the front panel output.
So if you plug a HDA front panel into an AC97 motherboard you will get no output on the rear audio output. Most users aren't going to find that acceptable.
If you plug an AC97 front panel into a HDA motherboard then plugin detection will not work correctly, it may detect devices as not present when they are present and vice-versa and the exact results may depend on what audio is playing. On some motherboards it may be possible to disable the plug-in detection so they can work better (though still not perfectly) with an AC97 front panel.
Your particular front panel looks to me like it is primerally designed for HD audio motherboards but with a hack to give limited support for AC97 boards. There are loops of wire on the AC97 connectors which will make the rear output work but it won't be switched by the front panel connectors like it would be with a proper AC97 front panel.
Intel introduced HDA back in 2004 though both AC97 and HDA were supported in paralell for a time but anything even remotely modern should be HDA. According to it's datasheet the ALC888 is a HDA chip.
I would suggest trying the HD audio connector first, if the rear audio output on your motherboard doesn't work try using the AC97 connector instead.
I'm not sure of the exact different (I assume that HD audio will gives better audio quality) but I'll always go for HD audio if the mainboard has it, if the mainboard only has AC97 then I'll use AC97 connector.
BTW, the HD audio connector can be used with AC97 as well. Incase your case only has HD connector but your board only has AC97