While Poisson regression assumes a (known) average, Negative Binomial regression is implemented using what is referred to as maximum likelihood estimation.

Remember that, although Poisson distribution assumes that the mean and variance are the same, sometimes data will show greater variability or extra variation that is greater than the mean. When this occurs, Negative Binomial regression is a better choice because of its greater flexibility in that regard.

To illustrate, what if we consider that a university wants to predict the average number of days a student athlete may miss each year. Predictors (of the number of days of absence from class) include the type of sport the student athlete is a member of and their average GPA score. The variable sport is a four-level nominal variable indicating which sport the athlete participates in (in this case it's either Football, Track, Field Hockey, or Volleyball).

If we profile our data, suppose we find the following statistics:

Football: M (SD) = 10.65 (8.20)
Track: M (SD) = 6.93 (7.45)
Field Hockey: M (SD) = 2.67 (3.73)
Volleyball: M (SD) = 1.67 (1.73)

We may look at the preceding statistics and see that the average numbers of days absent by sport seems to suggest that the variable sport is a good candidate for predicting the number of days absent because the mean value of the outcome appears to vary by the athlete chosen sport. However, looking at the standard deviations, we see that the variances within each type of sport are higher than the means within each level.

These are the conditional means and variances. These differences suggest that over-dispersion (greater variability) is present and that a Negative Binomial model would be perhaps more appropriate. You could still use Poisson regression, but the standard errors could be biased.

Negative Binomial regression takes advantage of one additional parameter (over Poisson regression) to fine-tune the variance independently from the mean (in this example it is the student athlete GPA score).