$e^{-5}$ error using Taylor's series
Use a spreadsheet and calculate $e^{-100}$ both ways. It will become totally obvious what is going on. "We avoid subtraction" is not the problem. The problem is that you are adding the sum or difference of large numbers, and each large number has a large rounding error. For one method, the rounding error is not large compared to the result. For the other matter, the rounding errors are huge compared to the result.