For what is the Thunderbolt AIC connector used?

Mainly, Thunderbolt consists of 4 PCI Express lanes (the PCIe version depends on the Thunderbolt version) and a DisplayPort connection. An Add-in card (AIC) gets these from the PCIe slot and the DisplayPort cable you plug into the card.

However, it is not enough to simply connect the cable to the PCI Express interfaces already present on your mainboard. Therefore, the card makes an additional connection, using the so-called GPIO (General-purpose I/O) header. Unfortunately, the exact nature of this connection is not documented. It is probably used by the chipset to dynamically reassign PCIe lanes as required.

The additional connector is required, the AIC will not work without it.

AIC = Add-In Card

You can transmit DisplayPort signals through Thunderbolt; more tidbits of information are at Can I connect a DisplayPort monitor to the Thunderbolt port on a Mac, and vice-versa?. For later versions of TB, you can daisy-chain displays, which could result in less clutter.

Asus call the AIC connector lead a "system-link cable", which leaves me none-the-wiser as to its purpose.

I do not know if you could operate a TB AIC without the DP input - I do not have the necessary parts to test that, and it could depend on if the driver insisted on having a DP input.

Here is an example of an Add-in-card:

http://www.gigabyte.com/products/product-page.aspx?pid=5210#ov

In the downloads tab/section there is a pdf that explains how it is plugged in to the 'Thunderbolt Header' on the motherboard, in addition to the PCIe x4 slot, as you suggested in your question.

The AIC has displayport sockets (input), and thunderbolt connectors (output). So I suppose it allows you to re-route an external (or internal) graphics card output to the displayport connectors, in addition to anything via the PCIe bus.