The Mitsubishi HC3800, as Mitsubishi's entry level 1080p projector has the luxury of a bigger budget. As it turns out, Mitsubishi has poured that extra budget into more performance, while keeping features very basic. In fact the Mitsubishi has less inputs than the HD20! They - Mitsubishi - might have at least sprung for a second HDMI input.
Both projectors are pretty limited in placement flexibility. That means ceiling mounting for most people doing a "permanent" installation.
The Optoma HD20 is nice looking, it's got a couple of curves. The Mitsubishi is more boxy, but with some sculpting, but nothing to write home about. The bigger difference is that the HD20 is white, and the Mitsubishi HC3800, a dark grey that might as well be black.
The HD20 and HC3800 have their control panels on the top, and their inputs (and outputs) located in the back. The Optoma projector has two HDMI 1.3 inputs, the industry standard. The HC3800, however has but one. For those doing switching through an AV receiver (with HDMI switchin), or routing everything through a PC, won't care. For some others it will be an annoyance, although one can buy a good little hdmi switcher for around $50 or even less. You'll need a couple extra short cables too.
When it comes to placement flexibility, let's call it a tie. For inputs though, the Optoma HD20 gets a win, besting the HC3800 by virtue of not only having a second HDMI input, but also a 12 volt screen trigger.
Below, first image, is the Mitsubishi HC3800 input panel, and below, the Optoma HD20's input panel.