Mickey's backyard BBQ is a country themed, crowd participation, dinner show kind of thing. Personally we like the HDD better, but we did have fun at the BBQ.
Your post says "you and your DH". I think the BBQ is more geared to kids. There is character interaction, line dancing lessons and sing alongs. There is corny humor, but not even close to the corny humor at the HDD. The food is basically the same as the HDD. The BBQ is outside under an open air pavilion kind of thing, so if it's hot out the only break you get is some fans that are set up.
My opinion would be if you've never done either the HDD or the BBQ, do the HDD and save the BBQ for later.