Non-overlapping confidence intervals is a much higher standard than zero not being in the confidence interval for the difference between two parameters. This applies all over in statistics, not just for this particular situation.
In this particular case, think about it this way: AUC1 is not in the confidence interval for AUC2, and AUC2 is not in the confidence interval for AUC1. Each of those confidence intervals is telling you that the other AUC is not a plausible value. Thus, the AUCs must be different, hence the small p-value when you formally test for equality.