If we dissect AIC formula by Akaike (1974), $$ AIC=-2lnL+2k, $$ where $k$ is the number of parameters, the two differences between $AIC_1$ and $AIC_2$ are the value of $k$ (negative binomial likelihood will have an extra parameter) and the $lnL$ values. Both are discrete regression models and they are often used to compare two or more models. However, one may argue if the difference in $AIC_1$ and $AIC_2$ are significant enough to conclude one model is better than the other. In this case, one may employ a likelihood ratio test and decide which model is the best.

See Likelihood Ratio Test for Poisson vs NB GLM

If we dissect AIC formula by Akaike (1974), $$ AIC=-2\ln(L)+2k, $$ where $k$ is the number of parameters, the two differences between $AIC_1$ and $AIC_2$ are the value of $k$ (negative binomial likelihood will have an extra parameter) and the $\ln(L)$ values. Both are discrete regression models and they are often used to compare two or more models. However, one may argue if the difference in $AIC_1$ and $AIC_2$ are significant enough to conclude one model is better than the other. In this case, one may employ a likelihood ratio test and decide which model is the best.

See Likelihood Ratio Test for Poisson vs NB GLM