Some of the differences between P&Co and C&J BG include channeled soles and the edge of the sole by the arch is curved (sorry don't know the technical term for this). These are extra detailing that some people appreciate but others might think negligible. IMHO I think the leather of HG is much better than that of the P&Co. HG I think has oak bark tanned soles which I am not sure if P&Co. have the same standard.
The difference between C&J and AE is there but I think it comes down to the fit of the last, style, and if you are willing to pay the extra $ for the C&J. I am still a firm believer that AE makes an excellent shoe at their price point.