Could be a time out with the "I'm OK" signal. With the Power on after Power off the returned signal may be faster as some capacitors are still charged.
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The ATX specification requires that the power-good signal ("PWR_OK") go high no sooner than 100 ms after the power rails have stabilized, and remain high for 16 ms after loss of AC power, and fall (to less than 0.4 V) at least 1 ms before the power rails fall out of specification (to 95% of their nominal value).
Cheaper and/or lower quality power supplies do not follow the ATX specification of a separate monitoring circuit; they instead wire the power good output to one of the 5 V lines. This means the processor will never reset given bad power unless the 5 V line drops low enough to turn off the trigger, which could be too low for proper operation.
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