What are the room dimensions and your seating distance?
I have a wall mounted 50" plasma and a 120" motorized screen in the same bat-cave both work equally well
the TV looks better with a couple small lamps on in the room, the PJ looks AMAZING for HD movies with all lights off.
10'-6" distance from my head to screen (tons of pic's in the link in my sig comparing both)
The flatscreen works best in moderate light situations where activity is going on during the viewing. Family members can read, do homework or even crafts during the show. This is the correct solution if you are looking for an enhanced TV experience, or in many cases for casual sports viewing. The sports guy can also put up multiple displays and watch a couple of games at the same time.
The front projector, OTOH creates a "theater". You sit down, turn off the lights, shut up and watch the movie. Many owners pride themselves in building a better theater experience than they get at the local multiplex. The downside is that you watch what is on. You don't knit, read, browse because it disturbs the rest of the group and usually requires a light level that hurts contrast. In addition, sound is an important part of the 'real theater' experience and is usually run at a level that discourages casual conversation.
My thought is that if you have to ask, the right answer is likely the flat screen.
Any recommendations on the best way to provide indirect lighting in a media room to combat this? I'm leaning towards these "eyeball lights" that mount on the ceiling but I know so little about this topic I could be easily persuaded that other types are better.
First, a big +1 on jayn_j's post - that is the best summary I've read here on the distinction between these two use cases...
Real theaters are "dark" because the technology required it, and the lack of light forces the social behavior of the group (well, it *should*).
I don't watch my PJ in complete darkness (quite dim, though), but you won't get fatigue from a PJ image because there's not "too much" light coming off the screen. Put a big flat-panel on high brightness and you're really spraying a lot of light into the room. You don't need to dim the lights at all for a flat screen except to provide the social cues (and avoid reflections if that's a problem).
Anything that doesn't throw light directly on the projection screen will work. Downlights work fine, eyeballs may be helpful in some areas, but aren't required. Avoid placing anything not on it's own (separate) dimmer control anywhere near the screen.