I find stair running doesnt hit the hams/glutes as well as hill running. I find you hit your quads a little more on stairs but have less “clawing back action” compared to hill running.
You are also limited in stride mechanics (particularly stride length and frequency due to the fixed height, depth and distance of the stairs)
That is just my personal experience though 
I look at a field I see grass
I look at a hill I see a training session
I find that my quads burn out doing stairs and really keep me from going while hills let me power throughout and I notice I get really good knee drive plus they feel a lot better to do while working me out harder.
My personal experience 
Chris is talking about the difference in surface - i.e. grass vs concrete, not the actual mechanics of sprinting
Edit made 
Not only surface but slope as well
Cheers,
Chris