Aside from the distribution, you could create a variable called $\texttt{nightevent}$, which is set to 1 if an event happened at night and zero otherwise. Next, create a variable $\texttt{southern}$, which is set to 1 if the record is for a southern state, or zero if from a northern state. Then regress $\texttt{nightevent}$ as the dependent variable on $\texttt{southern}$, which is the predictor variable. If the regression coefficient is negative for $\texttt{southern}$ it means southern states are more associated with daily events and northern are more associated with night events. Otherwise a positive coefficient would imply that southern states are more associated with night events and northern associated with day events. If the coefficient for $\texttt{southern}$ is significant, it means the test is significant.
Regarding the distribution, yes, binomial is appropriate, so you could use logistic regression to meet this assumption of a binomially-distributed outcome, i.e. $y=0$ or $y=1$.
The above is not appropriate for evaluating "what time" the events occurred, since that would entail use of longitudinal (time-dependent) regression.