I have been to both and personally prefer Chicago Cut. The menu is more robust, the food is excellent and the service was excellent. The atmosphere is steakhouse, but a little more modern/luxurious to me than DBP. There are great views (and a patio) of the River and the Loop.
I'm not the kind of person that just goes for steak + starch, so I think the menu is a HUGE plus for me...
Perhaps the one time at DBP was a fluke, but I've now been back to Chicago Cut several times and each has been really great.