Mickeys BBQ is great for interaction with the characters. We took my kids when they were 2 & 4 and they had a blast. Seating is picnic tables, rows and rows of picnic tables. It's outdoors so it is hot & humid.
HDDR is indoors individual tables/chairs, air conditioned etc. My kids have always loved the show.
Food is technically the same food, but I find the food at HDDR to be much better. It is fresher, brought to your table vs having to go get it. It also seemed that at the BBQ the food wasn't out very long. I'm going to guess because of the heat/bugs.
We totally enjoyed the BBQ for the characters my kids had a ball. None of us enjoyed the food. We've only done it once. HDDR everyone has always enjoyed it, we've done it many, many, many times.