My question is
$$\sum_{n=1}^{\infty}\frac{2}{n!}= 2e$$
and $$\sum_{n=1}^{\infty}\frac{n^2}{n!}= 2e$$
But each term in the series
$$\sum_{n=1}^{\infty}\frac{n^2}{n!}$$
except the first one is greater than each term in
$$\sum_{n=1}^{\infty}\frac{2}{n!}$$
So why isn't that
$$\sum_{n=1}^{\infty}\frac{n^2}{n!} > \sum_{n=1}^{\infty}\frac{2}{n!}$$
Please do note the comments noting that your first summation should start at zero. Anyway, you've answered your own question: it so happens that the fact that the first series has a larger first term makes up for the fact that every term in the second series except the first is larger than the respective term in the first series.
This fact should not be too difficult to swallow because the first few terms of both series are the ones that dominate the series, so you can imagine that the infinitely many terms that the second series is larger in each make minute contributions to even out the disparity due to the larger first few terms in the first series.