They have a simpler mechanism inside. If we're talking about OM Rugers, Colts, and Colt clones, the trigger pull is better from the start and also easier to improve. (Less so for New Models.) I find them much easier to clean. I like their traditional lines, the way they fit my hand, and how they point.
There are only a few reasons you could argue for a DA's superiority: It's faster to shoot, and it's faster to load. But I can't shoot a DA very fast anyway, at least not if I want to hit anything. I could get better with practice, but I could also get a lot faster shooting a SA. (Watch someone who's really good at Cowboy Fast Draw!) I find that I seldom shoot my DAs in double action mode. I usually thumb cock the trigger, same as I would a SA. As for loading, I think it's only significantly faster if you're using speed loaders or moon clips. I don't have either one, so I load them one at a time, same as my SAs. I think if I were to time a reload in one of my DAs I'd find it takes me about 30 seconds. With one of my SAs it's probably a minute or so. To me, that 45 second difference just isn't critical--I doubt I'll ever miss it.