For me it's the impact. At my size it is just WAY too much impact on my lower half.
I realize it's probably because I have terrible running form, and if I learned to run properly it wouldn't be as hard on my body. But I hate it, so I'm not motivated to learn to do it better.
The impact is eventually not good for anybody, no matter how good their form is or how good their joints start out. The chances that you endure knee problems you wouldn't otherwise have had are pretty high.
Most of the benefits average people are going for can be achieved with a brisk walking pace. And when most people say they run, they mean jog, particularly over several miles. Sprinting is not an option for most people for long distances, but in short bursts, even that's better for your joints than jogging.
That said, I'm no jogging Nazi. I don't go around telling people they should either walk or sprint. If the subject comes up, I'll say what I just said in case they care. Most people don't, and I don't blame them.
I could hardly ever get into running. I ran like a mofo for hours at a time on a basketball court. But I had terrible discipline and will power to go running. It just bored me to tears, I think. Unfortunately, everything like that has been off the table for ~17 years, when I screwed up my ankles and feet. I do well to ride my little stationary bike these days.