The additional 12V connector provides localized power for the CPU, and I believe, the AGP card. Not plugging this in results in a higher resistive path through the older motherboard power connector and the motherboard PCB resulting in a voltage drop (V = IR). The older motherboards could afford not to have this additional connector because currents were relatively low compared to modern AGP and CPU loads (so "R" was constant, but "I" was low, so Vdrop was low). In addition, new CPU's and AGP cards are using lower voltages than previously, so as a percentage a given voltage drop (say, 200mV) at 1.5V is of higher significance than that same drop at 3V.
But more to the question, would this cause damage? It could, but it probably wouldn't. It's not too hard to imagine a case where the load on the output of the motherboard voltage regulator is so high that the regulator has problems meeting demand at the lower voltage caused by the voltage drop and ends up causing voltage fluctuations. But I wouldn't think that this would happen due to careful engineering. I would imagine the biggest reason not to do this is that you could end up later having sporatic errors under high load situations and not know the cause of the problem.
Still there's a reason that it's there, and I can't see why you would want to risk data - or your electronics - over a couple of dollars. There are 4-pin molex to 4-pin square connector converters out there that are really cheap. Why not just use one of those to be safe?
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