Poisson distribution:

v^k e^-v / k! where v is the mean(=variance, too) and k is the particular outcome.

Gaussian with the same mean and variance:

Gaussian and Poisson for mean = v = 1:

Note how the gaussian continues to have values when its argument is < 0, whereas the poisson stops at 0. This causes problems with the gaussian because the number of counts never goes below 0 in our case. If we integrate from 0 to ¥, we miss some of the area under the gaussian curve. Also, the shape of the distribution is quite different.

Gaussian and Poisson for v = 5:

Gaussian and Poisson for v = 10:

Here, the shapes of the two distributions are essentially indistinguishable and the amount of the gaussian below zero is negligible.

Here, the shapes of the two distributions are quite similar and the amount of the gaussian below zero is very small.