Other than that might be taken for when typed as text, I'm not sure there is a difference.
Their graphs are different, that is why I ask. Here is how I put them into my calculator:
y = e^(x)^2 *this one looks like your run of the mill exponential compressed vertically
y = e^(x^2) *this one looks like a vertically compressed parabola, vertex @ (0,1)
I just need to know because I'm pretty sure I'll see it tomorrow on my final in the volumes of rotation section