What normally happens when the government gives out permits (or other benefits to producers) for free instead of charging for them? In a competitive market, the benefit goes to the consumers in the form of lower retail prices. The value of the handout to producers is competed away. Because the price to consumers is reduced, the quantity purchased increases.
But I don't think that can happen here because the quantity of permits is fixed. And therefore the quantity of goods that can be produced is fixed. Since the quantity of goods purchased cannot increase, the price of goods will not be reduced. Thus the handout to producers is not competed away after all; it is enjoyed by the producers. (Just like the gas tax holiday would benefit producers rather than consumers.)
Thus the handouts would create economic rents -- i.e., producers would enjoy supercompetitive profits.
This isn't really true, because the permits aren't "handouts." Under the status quo, energy companies can pollute costlessly. A cap and trade system restricts their ability to pollute, whether you give permits away or sell them. Either way, there's now an opportunity cost associated with pollution: either you have to buy a permit from the government, you have to buy a permit from another firm, or your forgo the opportunity to sell a permit on the open market. Any cap and trade system takes a resource that previously had a price of zero (pollution rights) and drives its price up to its socially efficient level, assuming the program is implemented correctly. That happens regardless of how permits are allocated. You could auction them give them away; the market price ends up being the same.
I'm sure Mankiw knows this. I think when he talks about an auction system as being the same as a Pigouvian tax, he means that both achieve social efficiency
and they both generate an identical amount of revenue for the government. Giving permits away achieves social efficiency too, but doesn't generate any tax revenue.
FWIW, the original SO2 program involved only a very small number of permits being auctioned. I think something like 95% of them were just given away to firms based on their previous production patterns. That program worked pretty well, and didn't generate any super-normal profits, because they made it costlier to produce SO2, not cheaper.