We have been to and enjoyed both, but to answer your question, we've been to Mickey's BBQ once and have reservations for the HDD for the 7th time at the end of March.
The backyard BBQ was fun, but it was hot (we went in July), it was mostly Country and Western themed (no offense, but not our cup of tea) and the humor was not as good as the HDD. But again, we did have a good time. There was a lot of character interaction for the kids and the food was good (although it's about the same as the HDD).
Now the HDD. Would not miss it when going to WDW with the kids. The jokes never change but never get old. And it's air-conditioned.