All of our horses are double registered, however I enjoy the AMHR over the AMHA.
The showing rules are typically the same in both, with some exceptions.
For example, a difference is in the Jumper class.
In AMHA the Jumper class is not timed at all and allows horses to come to a complete walk if its needed to come around the turns. As long as they jump the course clean it doesnt matter what gait they demonstrate while doing it. The jumps go up very high by the end and the horse that completes the course clean wins.
In AMHR, the first round the jumps arent very tall and it is not timed. You just have to go clean. Once your horse does a clean round you compete in the second round. It is based on speed AND cleaness. So, typically the fastest horse wins. But, even if the horse is the fastest it can get "faults" for knocking down rails.
It really just depends on personal preference. Both the World show and the National show are equally competitive in the showring. And, they both display BEAUTIFUL horses. However, the shows have completely different enviroments outside the ring. AMHR seems much more relaxed outside of the showring, where as AMHA seems to have the competitive tension the whole time--in or out of the ring. That's just my opinion about the two. I think everyone has their own favorite, it's all in the eye of the beholder.
Good luck, and hope to see you at the shows -- whichever one you choose!