Given how sparingly little we know about the Upside Down, it's quicker to list the similarities:
They're both dark, shadowy reflections of the "normal" world, and places of death and decay.
Beyond that if we want to compare them more deeply, the answers tend to be "we don't really know", so there's not much comparison to be had. After all,
we were only there for a few minutes of screen time total, and we saw all of the thematically important things: rot, fungus, weird stuff growing over everything, decay, death, etc. We also only saw one inhabitant: the thing the kids called the Demogorgon. There's a lot of suggestion at there being much, much more, but really the Upside Down we saw was pretty simple in terms of components.
There's one clear difference though:
Unlike the Shadowfell, which is quite significantly departed from the shape of the Material Plane, as best we can tell it seems the Upside Down extremely closely mirrors everything in our world, down to the placement of individual lawn chairs. It must look pretty strange in there when we're busy building something.
We might be able to do better comparison once there's a second season, but until then there's not much to be done.