I’m no doctor but that’s never stopped me from making a guess. ISTM that going up uses more of your glutes than going down does. Both directions use your quads, definitely, but for glutes they’re used more going up than down.
(Reminds me of the Seinfeld episode, ASSMAN.) 
I’m actually going to disagree with many of the posters above. I don’t think it’s possible to run down stairs at all (at least stairs with a typical rise).
Hear me out.
Running involves pushing off the leading foot and catching yourself with the trailing foot. You can run down a decline just by leaning forwards.
But leaning forward doesn’t really work going down steps, because if you leave your leading foot on the same step for too long the lower steps will be in the way when you want to bring the leg forward, and the step above also blocks the leg from extending out backwards.
So how come some people (including me) can apparently “run” down stairs? Because that’s not really running, it’s more like making small jumps with alternate feet. Which is something most people aren’t used to doing. And is a little risky.
In my younger days while road-tripping with my parents and siblings, we stopped at a rest area to make sandwiches and have lunch. I was about 18 years old. Having a few minutes with nothing to do, there was a nearby rock wall, nearly vertical and about 25 feet high. The rocks were big, about beach ball size and bigger, and the wall to my untrained eye offered many footholds and handholds. So I decided to climb up a bit.
It was fun and a little challenging, and pretty soon I found myself 15 feet up. The climb up was fairly straightforward, with some challenging parts, but not too bad.
Being 15 feet up, I did not want to climb much higher, in case I fell. I didn’t want to get too badly hurt. So I decided to climb down.
I was surprised to find that the climb down was much more challenging than the climb up. What had been a fairly straightforward climb up now took maybe 4x the time to get down. I eventually made it down but to my young brain I was stymied. Why the big disparity?
Once on the ground, lunch was ready and we were called back to the car and picnic table. But I stopped for a few minutes to study the wall and figure out… why???
That’s when I realized it’s all about your eyes. Your eyes are perfectly positioned to study your handholds and footholds when climbing up. But when going down your eyes are far from the footholds you will need to use, and your view of them isn’t ideal — they’re seen from above instead of from the same height, and it’s harder to judge their suitability (will it be sturdy? is it big enough for my foot?).
A nearly-vertical rock is more extreme than a set of stairs, but in the continuum between a flat sidewalk and a vertical rock wall you have all sorts of inclined surfaces, from mild ramps and mild stairs, to steeper ramps / stairs / paths, to severe stairs / ladders / trees / hills, etcetera. But the limitation applies to them all —
It’s all about your eyes.