I once took my shop owner's bike out for a couple hours, and that was the only time I've tried carbon bars. Personally, I didn't notice a huge difference in damping, but it was only one ride. I just think the marginal weight savings of carbon bars over aluminum bars doesn't outweigh the extra durability that aluminum provides.

On a side note, I've also rode carbon seatposts, and I never noticed a huge difference. I'll take a nice aluminum seatpost (i.e. Thomson) over a carbon one anyday.