I recently messed up a stock installation of an Athlon64 x2 HSF. I had aligned the HSF to some components on the motherboard but realized after planting it down that part of the CPU heatspreader wasn't even covered. I took off the HSF (had to pull slightly to unstick the thermal paste, part of which transferred to the CPU spreader) and then aligned it properly and planted it down again, this time fitting the screws in.
Under Core Temp, idle temperatures of both cores looked fine and within 1-2C of each other. During load, one core would reach 58 while the other remained at 53. Suspicious, I took the case apart and reinstalled the HSF, this time cleaning both surfaces and putting on AS5 (which looked the same to me as the stock paste - opaque gray and highly viscous). Afterward, Core Temp reported 48-50C for both cores under load.
Based on this, I've come to think that in cases where the cores don't reach the same temperature under load, there's likely a problem with the HSF installation. Perhaps there was too much paste on one side... or perhaps the screws on one side are looser than on the other. This probably won't break a stock processor, but I'd think it would significantly affect one's overclocking success.