Data inspection and modeling for counts, Part 1

The answer would depend on various details. If you're looking to compare groups for a difference in counts between them, then general Poisson regression would work. Just use the group membership as a factor in the model.

However, if you want to characterize the statistics of the difference itself, then yes, some other approach would be needed. The possibilities depend on the details. If it's a change over time, then a time series regression could work. If it's about the absolute difference in itself then I would probably lean towards bootstrapping because it's hard to say much a priori about the distribution or expectation.

A way to sidestep some of those issues is to consider whether proportions might work for your problem instead of counts. Then an option like beta regression could work (where the outcome is a beta variable, i.e., a proportion).

For my 0.02, those are probably the order I'd try: regression with a class variable; time series if appropriate; bootstrapping; beta or similar regression. HTH!