This is too easy. Just think of the HC3800 as a "poor man's" BenQ W6000. The HC3800 is barely more than half the price. It lacks a dynamic iris (the W6000 has one - though not the smoothest), and overall the W6000 is a real step up in black level performance.
Both are very good at having that DLP look and feel, with rich deep colors, and very good skin tones. The two projectors side by side, are basically very similar in terms of screen image, but the BenQ W6000 definitely has the advantage. Both are very bright in "best" mode, and above average in brightness in brightest mode, but, again, the BenQ W6000 has a significant advantage.
The BenQ W6000, however comes with only a one year warranty.
Consider the W6000 to the the step up DLP projector for those willing to spend $1000+ more. As nice as the HC3800 is, the BenQ is worth the difference. I can't think of any reasons, other than cost, or warranty that favor the Mitsubishi. They are two similar projectors overall, one simply is a more expensive unit with more performance
Mitsubishi HC3800 vs. Sharp XV-Z15000
This is a tough one. I really would have liked to have had both projectors here for a side by side comparsion. Again, we are looking at two DLP projectors, both with especially good color and skin tones.
The Sharp XV-Z15000 definitely has the advantage in black levels. It does particularly well in this regard. While I can't be certain, it's likely that if you turned off the Sharp's dynamic iris, the Sharp and the HC3800 would have similar black level performance. But, the Sharp does have the dynamic iris, and the better black levels.
There are lots of trade-offs though. The Mitsubishi HC3800 has the sharper image of the two, or rather, they are similar at the point where you focus, but the further you move from that point, the more difference. The Sharp's sharpness, (did I really write that), rolls off faster to the sides and corners. Overall, the HC3800 will look a touch sharper.
Currently there seems to be about a $500 - $600 price differential.
HC3800 vs. Optoma HD20
This is perhaps the comparison of interest to the most people. Both are DLP 1080p projectors, but the Optoma is down at $999, and everyone seems to want to know if the HC3800 is worth the difference. My take is yes, if you've got the bucks, go for it. The Optoma does just fine, but isn't as refined. First of all it's a noisy projector, even compared tot he HC3800 which isn't particularly quiet. It also seems to throw a lot of heat (it's very small even compared to the HC3800). It also leaks light out the front, although not enough to be a big problem. Black levels are not as good as the HC3800 either, but the HD20 is very good on shadow details, at least as good as the HC3800.
Ultimately, picture quality is similar, but the HC3800 has the advantage, thanks to the black levels. The Mitsubishi has more placement flexibility