In modeling claim count data in an insurance environment, I began with Poisson but then noticed overdispersion. A Quasi-Poisson better modeled the greater mean-variance relationship than the basic Poisson, but I noticed that the coefficients were identical in both Poisson and Quasi-Poisson models. 

Is this an error? If it is an error, how might I resolve it? If it isn't an error, why might this be the result and what would be the benefit of using Quasi-Poisson over Poisson?

Things to note:

 - I am using glm() with family = poisson or quasipoisson and a log link
   for code. 
 - I tested for overdispersion via dispersiontest in the AER
   package. My dispersion parameter was approximately 8.4, with p-value
   at the 10^-16 magnitude.  
 - When running the Poisson code, I come out
   with warnings of "In dpois(y, mu, log = TRUE) : non-integer x = ...".
 - The underlying losses are on an excess basis, which (I believe) prevented the Tweedie from working - but it was the first distribution I tried. I also examined NB, ZIP, ZINB, and Hurdle models, but still found the Quasi-Poisson provided the best fit.