Brian,
For my own education, can you describe more exactly what a saturated inductor "sounds" like?
I think Karsten now has the most practical experience to answer your question, but I will try and approach it from a somewhat "theoretical" standpoint.
First of all, the magnetic behavior of core materials (other than air) can be quite complex. In fact, this is often the reason many suggest for sticking with air-core coils. But...there's always a catch. In woofer circuits (in particular) we usually don't want to insert any series resistance. Typically, the rated sensitivity of a speaker is based on the sensitivity of the woofer(s) used. Any added series resistance in the woofer circuit will have the effect of doing 2 things:
1) It will waste power and consequently lower the woofer's voltage sensitivity. That's not good as we all like our speakers to be as sensitive as possible - unless we happen to have unlimited amplifier power at our disposal.
2) Added resistance lowers amplifier damping. That's not good either if you like fast, clean and powerful bass response. An old trick to get more bass out of a speaker is to intentionally add a small amount of resistance (i.e, as in a small value series resistor). Typically the values used are 0.5-Ohms or less. This lowers amplifier damping and allows the woofer to "resonate" more at the box tuning frequency. It works, as long as you don't mind the transient response getting a little sloppy.
Well...back to the point. An air-core coil will have considerably more resistance than a iron (or whatever) type core - for a given physical size and inductance value. This is because without the benefit of a magnetic flux concentrating core, an air-core has to have a lot more turns of wire to achieve the same value. The only way to reduce the resistance is to use very large gauge wire. If you do that, the coil can become enormous in size and very expensive. Have you checked out the price of copper lately?
So, to get a large value inductor with low resistance in anything resembling a reasonable size and cost, we're usually forced to go with some form of magnetic core material. And that can be fraught with hazards. Yet, done correctly its actually a superior way to go over all. A well done design can provide extremely low series resistance in a manageable size and at reasonable cost.
But magnetic materials other than air have "personalities" that can vary widely. It all has to do with an ugly little property we refer to as "hysteresis." Think of it as a form of memory. Once the core has a magnetic flux established, it doesn't want to "let go" once the field tries to change polarity. Well, audio signals are "AC" and change polarity every half-cycle. If the hysteresis is significant, distortion results.
Therefore, most core materials chosen for low to mid audio frequencies use a material that has the lowest hysteresis possible. That material is usually a form of "silicon steel" (steel with silicon alloyed in it - duh). The problem is that the good stuff ain't cheap and the cheap stuff "saturates" easily (and has inferior hysteresis performance).
What's saturation? It's when the core material is so full of magnetic flux lines that it can't permit any more to form in it. The more current flow to the woofer, the more flux lines are needed inside the core. Once it hits it's limit - that's it - distortion. It's a lot like when an amplifier clips. There just ain't no more signal going through. But a saturated inductor core can be even more insidious.
Once the core saturates the inductor can looses virtually all of its "inductance." In a bad case all that remains s what we call "leakage inductance." That ain't much either. The result is as if the inductor was removed from the circuit or bypassed with a straight piece of wire. Now your "Low Pass" circuit ain't "Low Passing" anymore - it's "All Passing." That means high frequencies can get to the woofer now.
But core saturation only really occurs at the signal peaks (unless its REALLY saturated). That means at those moments all kinds of hash can get through to the woofer. Some of that hash would be high frequencies from the amplifier (that should have been blocked by the inductor) and some would be from the inductor itself. You see, when an inductor (or transformer) saturates, it
generates all kinds of harmonic distortion.
Then we have even more uglies to add to the mix. The back emf of the distortion products generated by the saturated inductor are "reflected" back to the amplifier. If the amp uses negative feedback, it will try to correct for this. Some of the distortion it will get rid of, some it won't and because of any phase errors in the feedback loop, the amp will generate distortion products of its own and send them right back down to the speaker. Such a back emf would be quite complex - both from a frequency and dynamic standpoint - and well outside what the amplifier would "expect" to see. We sure can't blame it if it can't handle the whole mess.
And guess who's sitting there while all of this is going on? Your beloved little tweeter - that's who. Then of course, since many of these distortion products will lie in the tweeter's operating band, the High Pass circuit will more than oblige and drive them right down that poor little tweeter's throat. Now you can hear them in all their glory. What these nasty things will do to soundstage, imaging, detail, resolution - you name it...is anybody's guess. One thing for certain though, you can bet it ain't good.

So...what does it sound like. Well, I took the long way to get there but I'd say... a
BIG, WET, FART! Karsten has posted more above on the subjective effects, so I'll leave it at that. With regards to:
One strange thing, which I can't explain (maybe Bob can) is that the upgrade also has a substantial impact even at relatively low listening levels.
Karsten is partially correct in his assessment that there's still pretty big transients coming through, even at low volumes. Above and beyond that I would suspect that the old inductor's core had a poor hysteresis response. That would manifest at any level, regardless of amplitude. Cheap steel sounds bad at any volume.
Dissertation complete. I'll go quietly now.
-Bob