From a recent email exchange:
I agree that you should never compare p-values directly. The p-value is a strange nonlinear transformation of data that is only interpretable under the null hypothesis. Once you abandon the null (as we do when we observe something with a very low p-value), the p-value itself becomes irrelevant. To put it another way, the p-value is a measure of evidence, it is not an estimate of effect size (as it is often treated, with the idea that a p=.001 effect is larger than a p=.01 effect, etc). Even conditional on sample size, the p-value is not a measure of effect size.