With regard to the following:

 - the mean of a Binomial dist is $np$
   
 - the variance is $np(1-p)$
   
 - the mean of a Poisson dist is $\lambda$, which we can imagine as $n\times p$
   
 - the variance of a Poisson is the same as the mean

Now, if a Poisson is the limit to a Binomial with parameters $n$ and $p$, such that $n$ increases to infinity and $p$ decreases to zero while their product remains constant, then assuming that $n$ and $p$ are not converged to their respective limits, the expression $np$ is always greater than $np(1-p)$, therefore the variance of Binomial is less than that of Poisson. That would imply that the Binomial is below in the tails and above elsewhere.