If this is an abstract question, I'd say that nothing changes. If the two PSUs have the same components and cooling mechanism, since the power drawn is the same, I expect the heat produced would be the same, so the fan should spin at the same speed on the two PSUs.
Also, consider that a PSU has its maximum efficiency when it is run at about 60-80% of its maximum and the efficiency diminishes with load. This means that more energy is lost in the circuitry (in form of heat). A 1200 W psu would be tuned for maximum efficiency at a higher power, so it would be less efficient than a 600 W PSU at 400 W.
In a real scenario, I'd also take into account that a 1200 W PSU might have better components than a 600 W PSU from the same manufacturer, to ensure better stability at higher power loads. This could bring better overall efficiency and reduce the heat produced, and thus the fan speed.
To have a more experimental answer to the question, you can read these two reviews of the same PSU model (550 W and 1200 W) performed by the same website. Unfortunately the fan noise is below the least audible level for both the PSUs, but you can see that the fan noise only increases when the 1200 W PSU is loaded with more than 900 W.
4There is no telling which one is quieter. For all we know the 1200 Watt PSU has two fans and the second one only kick in under high load. (The first fan could be identical to the fan in the 600 W version). -- TL;dr. There is not one single clear answer. – Hennes – 2013-03-28T15:54:44.807
Yes, but my theory is that manufacturers has logic that sets RPM at a certain load level, so 1200W at 33% would be spinning slower than the 600W at 66% load, and usually slower RPM will result in less noise. Though I don't know if this theory is true in practice. – KoKo – 2013-03-28T15:57:54.827
It could well be true for many, or even most PSU's. But there is no sane way to compare any 600W PSU with any 1200 Watt PSU and make a safe prediction. You will have to compare specific models. – Hennes – 2013-03-28T16:03:00.327