I can see the difference when projecting a computer screen (e.g., Windows desktop, start menu, etc) in a 800x600 Vs. 1280x800 resolution for example. But when watching movies is there going to be a difference? It seems you will not lose any part of the picture regardless of resolution. Is this correct?
Of course it matters. Even at smaller projected sizes, the resolution makes a huge difference.
HD video is between 1 and 2 million pixels, if the projector cannot display all of them, there will be a loss in video quality. Additionally, depending on the distance from the image and the size of the image, you may be actually able to see the pixels on the projection location. A 1280x800 projector will display all the pixels of 720p video. An 800x600 projector will display less than half of them.
At a distance of 20ft, I can easily see the difference between 720p and 1080p on a 50" display, and on a large projection surface (say 150"+), I can see the individual pixels of a 1080p projector. More pixels is better.
That is correct. The picture will be scaled to fit in a given resolution.
It might not be the same picture though. Like when the movie's resolution is greater than the projector's. It makes sense that detail, or call it resolution, will be lost.
Using a 1280x720 resolution movie as an example, a projector set at 800x600 will have to throw away a lot of information, since it's new job is to display 921,600 pixels (1280•720) with less than half that available to it. The 1280x800 setting can comfortably fit everything in its workspace.
Still though, for projecting movies, the thing that matters is up to you when you're sitting in your seat. If you have a 4:3 screen that can only accommodate 1280x800 by shrinking the image far beyond the size of how 800x600 compensates with the movie, then maybe 800x600 is better.
But if you're talking about details, and not cropping, it matters a lot.
The most basic concept in resolution is that the further you are away from the projector, the less the pixel density. Let's say that your projector lense is 4" in diameter. If you focus on a screen that's 1 mile away, guess what you will see? If the number of lumins isn't beyond all reason, you won't see anything and your pixel density on the huge screen will be virtually less than 1 pixel give or take. Basic physics teaches about focal points and distance from source. I assume that my home projector has a much better pixel density than what I see at a movie theatre for 1 reason. The typical distance from the projector in the theatre to its screen is over 70'. In my home theatre, we sit about 15'-16' away from the screen.