Difference between expanding $\frac{1}{1-x}$ around $x=1$ and $\frac{1}{1-e^x}$ around $x=0$

Question

The title pretty much gives it away already, I'm trying to find the difference between Laurent expanding $\frac{1}{1-x}$ around $x=1$ and $\frac{1}{1-e^x}$ around $x=0$. The first expansion does not work out very well, while the second one can be expanded quite easily. I find it odd that there is such a big difference between the two functions, and I was wondering if someone could give me some insights as to where this comes from.