Due to the subtractive cancellation in the expression for $f''$ there is a pronounced detoriation in accuracy as $h$ is made smaller and smaller.

It is instructive in this analysis to rewrite the numerator of the computed derivative as $$(f_h -f_0) +(f_{-h}-f_0)=(e^{x+h}-e^{x}) + (e^{x-h}-e^{x}),$$ as $$(f_h -f_0) +(f_{-h}-f_0)=e^x(e^{h}+e^{-h}-2),$$ since it is the difference $(e^{h}+e^{-h}-2)$ which causes the loss of precision.