What is the best (smallest) bound found so far for the error when using the Poisson approximation to the Binomial distribution? And for which values of $n$ and $p$ is the error smaller than the normal approximation?
I couldn't find anything online.
As a side note, I do have found a bound for the normal approximation (although I'm not sure it's the best)
Normal approximation bound I have found (accordind to this SE question : Berry-Esseen bound for binomial distribution)
$$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}, \quad C \leq .4215$$