big rig vs. the second system

I've been listening to a number of recordings recently on my primary system and then immediately afterwards on the secondary one, with some pretty consistent results.

The biggest difference that I notice is not in ability in the treble, mid-range or bass, not in timbral accuracy, not in imaging, not in soundstage, not in rhythm and timing, etc., but rather in a sonic characteristic that I'm not sure quite how to name. Presence--immediacy--shimmer, are the three words that I can think of for the moment that come closest. It's that you-are-in-the-same-room versus the you-are-in-the-next-room sensation. And although it's one of the hoariest audio-reviewing cliches out there, you could describe it as as the difference between a non-transparent acoustic veil being lowered and raised. Perhaps there's a solid reason why this is such a cliche.

I wonder if this is just my experience, or whether others find that this is the primary difference, too.
I think it would vary with which specific systems you are comparing... What you've described sounds like a difference in frequency response to me... It's probably related to the respective rooms and the particular speakers. One pair of speakers may have a small elevation in the midrange while the other speakers may be level or slightly reticent.

Anyway, that's the most likely explanation for what you've described (though not the only one).
Check out this response curve.

I am not sure what speaker you are comparing this too but the off axis response will have as much influence on what you hear as the on axis response when placed in a typical room. So if you can eyeball the average response then it will give you an idea of how the big rig will sound in comparison to others. Areas where the combined response is strong (such as in the bass) and between 4 to 7 Khz will stand out in the presentation. Presence is defined here.

Does this match or tally with your observations? - normally it should.
Stuffiness, congestion, lack of clarity or definition, veiled, not transparent. . .

We've all heard it, and too often. Sometimes this may be related to frequency response. Sometimes off-axis response, as Shadorne has noted. Another possibility is related to resonance. Undesirable resonances of the drivers, the cabinet, high Qs, and other sources. It's a very common problem, and a problem very readily exposed with full orchestra. Nothing will reveal this flaw faster or more obviously, especially when the cellos or basses are playing.
You guys may well be right. I don't know enough about psycho-acoustic phenomena to be able to confirm or refute. The problem, as I'm hearing it, may well only actually be over a narrow band, but the mild "veiling" effect that I'm talking about certainly *sounds* as if it extends from top to bottom.

The thing is, I wonder if I would even perceive it as a problem if I didn't have something else to compare it to.

In comparing the two systems, different equipment, different room, different everything, I know that there are so many variables that it's totally apples and oranges (well, more like apples and mangoes).

But what I was trying to get at had to do with how we, generally, make judgments about the musicality and clarity of systems. I have to wonder, as the quality of the components rises, and the synergy between them rises, if there isn't a point, an incremental leap, an elbow in the graph, where things rather suddenly get significantly better. But we need a good auditory memory to perceive it as such, if we don't have something immediately to hand to compare with. It *feels* as if the two systems I'm talking about are on either sides of that elbow, which BTW is fine, as the second system is purely for non-critical duty.

These are only musings, not conclusions. Perhaps I'm just reinventing the wheel. But I know what I heard--even if I have no firm idea why I heard it...
You might not have noticed without the A/B comparison, but now that you have, you've learned how to recognize it without the comparison. These kind of broad, general characteristics can be remembered fairly easily, even though finer specifics are poorly recalled.

Different rooms certainly have an effect. Voices sound different in different rooms, sound different on the phone or outside, but once you've learned the sound of someone's voice, you can be recognize anywhere. Speakers may sound better or worse in this room or that, but each speaker will still have its characteristic sound anywhere.
Without more information on what you are comparing - it is impossible to comment further.
I guess the scientific answer would be to temporalily install the main system where the secondary one is, and listen and compare. But as there are stairs involved, and some very heavy equipment, I don't think it's going to happen any time soon.