belts are subject to more debris damage than chains. I know a couple people who have been stranded by broken motorcycle belts, in both cases some road bit got between the belt and pulley, punched a hole and very shortly there after it snapped. Otherwise they are clean and low maintenance, rather pricey compared to chain though and a pain to change.
...and the first use of belts on bikes was over a hundred years ago, they may actually predate the use of chain drive.