Jeff.
I'm not an expert on projectors but am familiar with using them. From basic physics and optics:-
The size of the image on the screen at a particular distance is related to the lens focal length not the resolution. Just think of a camera working backwards, if you see what I mean.
A camera of a certain type with a higher resolution does not have a bigger sensor (inches or mm) than one with a lower resolution.

A higher resolution projector will give you more pixels for a particular size on the screen and therefore will always be "better".

Using a HDMI connection cannot not change the resolution of the projector, that depends in its "sensor" (not sure of its correct name) which is fixed. Think of a monitor with both types of connection.
There will be differences in the processing of the basic data from the input to the projector "sensor" and HDMI  (which is like DVI on a monitor with sound I understand,) and for monitors, DVI gives better results than VGA.
We have a policy/recommendation at our club that projected images are best sized to the projector resolution to avoid the projector software messing with the pixels - sound familiar to Qimage users?
Regarding the laptop and using a dual display setup, it's important that the number 2 display (projector) is set to the correct resolution. I need to check, but the laptop may have to use the same resolution as the projector. Probably depends on whether you are using a cloned or extended desktop. Something else I'll ask about. The graphics card must be capable of working at the appropriate resolutions too.

That's my understanding regarding your query but I'll be seeing a friend this evening who's the projector guru at my club, I'll talk to him too and pass on his wisdom.
Terry

Jeff,
I spoke to my friend and he tells me that most of what I said is more-or-less correct but read on.
My club is considering getting a new projector and rather than going up just one level to a 1400 pixel job, move up to  full HD equipment 1920x1080 pixels (1080p). So, that is a recommendation for future proofing. 

He thought there was not much point in having an HDMI connection unless you went above 1400 pixels wide.
The resolution of the host laptop can be less than the projector, providing it's graphics card is capable of the outputting the projector resolution.
In that case you would have to use an extended desktop and an application that is capable of having the window with the images dragged onto display 2, ie. the projector.
Not all applications can do this. MS Power Point is a dual monitor application, the software our club uses for competition is (forgotten it's name) and of course, Qimage can be used in this way in slide show mode. I've done it but on a  1220x800 laptop and a 1024x768 projector. It does not work with a 1080p TV because my laptop graphics card is not adequate.
It sounds as though you'll have to put your club fees up, new projector, new or modified laptop, big screen .... 
Terry

Hi Jeff,
You are probably right, most laptops have "on-board" graphics, ie. as part of the motherboard. It may be possible to have a separate graphics card added in some way, but I don't really know. If the laptop has an HDMI socket, then it's likely that the on-board hardware is ok for 1080p.
Terry