[AaronBrown]

Your question is not quite clear.

I think you mean you have two large POPULATIONS, rather than distributions, otherwise the large N does not make sense. You know the means and standard deviations of the populations. Or you might mean that you have two distributions, but the means and standard deviations are reliably estimated from large prior samples.

I then think you are selecting one item from each population independently and want to know the probability that that the item from the first distribution will be larger than the item from the second. But you could mean that you are selecting more than one item from each, in which case I assume you would compare the means of the samples from the two distributions.

The independence or not of the samples is critical to this problem.

If we select one item from each population independently, then we know X1 - X2 has mean M1 - M2 and standard deviation [S1^2 + S2^2]^0.5. This has nothing to do with Normality, it will be true for any two populations. Normality allows us to put a precise number on the probability that X1 - X2 > 0. However, even without that assumption we can set limits on the value.