The mean and variance of Poisson distribution are equal This blog claims that the fact that the mean and variance of Poisson distribution are equal can cause problem. Could you please elaborate why this is the limitation and how it can affect models?
 A: When mean and variance are equal, variance increases as mean increases.
Problem in fitting poisson GLM : Overdispersion

Many a time data admit more variability than expected under the assumed distribution. The greater variability than predicted by the generalized linear model random component reflects overdispersion.

Source : https://web.archive.org/web/20130621133920/https://onlinecourses.science.psu.edu/stat504/node/162
Why use poisson GLM :
In the case of linear models, sometimes you observe a small difference between fitted and actual values (desired) when fitted value is low and a large difference between fitted and actual values (not desired)  when fitted value is high. This is called heteroscedasticity giving a funnel shaped plot between residuals and fitted values - try plot(lm) function in R.
A: Just read the next two sentences:

There is no way to increase the variance without increasing the mean. Unfortunately, in many data sets the variance is larger than the mean.

If you model some phenomenon with a Poisson distribution, you are tacitly imposing this constraint that the mean and variance must be the same. If your real-life phenomenon does not exhibit this property, then it may not be a good idea to model it with the Poisson distribution.
