As I say repeatedly, take two very good projectors, and compare on bright, and mid-bright scenes, they will look very close to each other, even when considering black levels. Look at some mostly very dark images and especially those without any really bright areas (like the Bond "train" image I use in all reviews), however, and it's almost "night and day" - let's make that "night and dusk". In the case of these two projectors, it really is the black level performance that sets them apart.
Let's look at some side-by-side comparison images, as they relate to black level, and shadow detail performance. Note, the HD20 will be on the left in all of these images, and the HC3800 projector will be on the right.
All of these images show at least some of the difference, but the best of them is the third one down, an overexposed shot of the satellite in Space Cowboys.
If anything, the HD20 may have the slight advantage in shadow detail, however, I'll attribute that to the idea that mostly, the dark shadow detail is there on the HC3800, but because the same dark shadow detail will project with lower brightness (since the blacks are lower), that detail is harder to see on a projector with better blacks. What I'm trying to say, is, that at first glance, the HD20 may look like it's doing a better job at dark shadow detail, but in reality, they are comparable.
By any measure, the Optoma HD20 does a really very good job in terms of dark shadow detail. The HC3800, by comparison, hangs right in there, yet is still better than, say, the Epson Home Cinema 8100, with which it also competes.
Unfortunately I did not do any side by sides from Casino Royale, but here are the two projectors respective "train" scene images. The first is the HC3800, which is a bit more overexposed. That makes it a difficult to confirm what I have been saying.