How does the normal distribution differ form the Poisson distribution?

n $\rightarrow$ $\infty$ , p $\rightarrow$ 0 in Binomial b(n,p), the distribution becomes Poisson ( still discrete) whereas n $\rightarrow$ $\infty$and p$\rightarrow$ 1/2, b(n,p) tends to N(np,sqrt npq), which is continuous.