I am no audio expert but I think AC97 is Intel's audio codec (began in 1997 as the name implied) but was later (around 2004/2005?) superseded by Intel's HD Audio (High Definition).
In the days of 90's, people basically listened to stereo music (2-channel music CDs) and AC97 was designed for that. But as surround sound gains popularity (particularly in movie DVDs), AC97 couldnt keep up and so Intel used HD Audio to replace the AC97.
So, I suppose if you use only 2 front speakers, the two shouldnt have much difference. But if you have 5.1 or 7.1 multi-speakers setup, use the HD Audio.
Actually, that's incorrect. AC'97 is a product of Realtek, and is still manufactured with AC'97 being the name of the processing chip, but with monkiers like "HD" or "5.1" being added on when they upgrade the chip to incorporate high frequency (96/192kHz) processing, or 5.1 output respectively.
HD Audio is a product of Intel, which is going into direct competition with the AC'97 onboard audio solution. Intel wanted either to save money on their own motherboard manufacturing or they wanted another market they could make some sort of money out of and the onboard audio market was one that was relatively untapped by anyone other than Realtek.
So here we are, every non-intel mobo out there with AC'97 onboard audio, either HD or more commonly now, 5.1 version installed, or the intel mobos, with their own proprietary audio processing chip onboard.
The reason there are 2 different connectors, is each chip uses the same 2x5 pinout, however, the key (blank) is different and the output/input pins are also different, as Realtek would have probably tried to sue the pants off Intel with whatever they could. The key pin on Intel's HD audio is pin 2, Realtek's AC'97 is pin 4 (if my memory serves me correctly). And again, the pins and what they do are all different. So really, it's just 2 competing onboard chips, and it's up to if you have an intel board or not, and if you don't, then don't even look at the HD Audio plug, you more than likely have the Realtek chip, and need to use the AC'97 plug.
This plug and pinout is for the front audio, which typically is just Front L/R for plain stereo output, and a mic input (stereo pin connections are available, but typically needs to be selected as such in audio control).