I've driven, fairly extensively, both the MDX elite and an Explorer Platinum. I'd take the MDX without a second thought. Yes the Explorer has power, and definitely feels way quicker than the MDX, it's also a little more spacious. However, I have not been impressed by the build quality on the Explorer; the last tester I had, the thin metal (aluminum?) hood would visibly vibrate on the highway, it drove me nuts! Fuel economy in the Explorer is also pretty bad, especially in the city, I'd probably save a litre or two per 100kms in the MDX.
The MDX's big advantage in my eyes is that it's a much more confident drive; it feels smaller (the Explorer feels massive thanks to extremely deep a-pillars), handles sharper and is much lighter on it's feet, which is nice when you need to dodge the usual road hazard drivers.