The above difference is because the z transform is only valid for non-negative integers. In fact, if you plot the two functions h and h2 for integer values only, you will see that they are identical. Alternatively, if you look at the plot you posted, you will see that for integer values, the two curves have the same values.
So in summary, when you take the inverse z of the z transform, you are taking the inverse z of the coefficients of a series, rather than a continuous function. That's why the inverse z of the z transform is also only valid for integer values.