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.
