Use Negative Binomial or Poisson to handle counts data?

There is a simple regression algorithm for counts data, called [Poisson regression](https://en.wikipedia.org/wiki/Poisson_regression). This algorithm assumes that every regressor has a multiplicative effect. It's similar to computing the log of the data, except it works even when the data has zeroes.

It's conceivable that you could replace the Poisson distribution with the more general Negative Binomial distribution. The NB distribution is a generalization of the Poisson that allows the mean to be different from the variance. In contrast, in a Poisson distribution the mean is always the same as the variance.

The main difficulty with just changing the Normal distribution to a Negative Binomial is it's then necessary to add the constraint that $0 < \mu < \sigma^2$.

Does this seem like a good idea?