The filter is a dual-cartridge filter made by "pur" and works really well (similar to their tap-water filters I'm sure). I always use a decent car washing detergent anyway, never just "no soap" if that's what you mean (if you don't get the dirt off with soap you'll have spotting).
Their system is designed so that their soap sheets the water off. In fact it will start to lessen any "beading" effect your wax causes (some people prefer that, but their soap no doubt leaves a film of some kind to cause the effect). Your regular water is used in volume to wash the dirt and soap away, then their filtered water is used to rinse your "bad" water off the car and leave clean filtered water behind. If you don't completely wash the hard-water off the car you will have spots!
Now - their "sheeting" action soap helps in this regard in that the water tends to not bead up on the surface in big puddles, but some waxes/polishes (like NXT) really fight their soap and still bead so the no-spot effect isn't quite as dramatic. I found that when I was waxing the Miata with a real "wax" (I was using Pinnacle Souveran) it tended to sheet a TON more than when using NXT - and the no-spot effect was pretty dramatic between the two. The effect on rainX-ed glass is even moreso.
With all that having been said I also pointed out I used NXT car wash on the car with the filtered water and it worked. I've had "perfect" success, and I've had spots doing it that way. I think I'll assume when it was "perfect" I either still had some film on the car from the Mr.Clean soap, or the wax/polish was pretty well worn so I didn't have a really good beading action - but even so, with a fresh coat of NXT using NXT car wash I still had better luck with the filtered water than without.
So to summarize, I think you can see that their system is really depending on their soap for the "full effect". The average car that is outside everyday and waxed every 6 months will probably have a reasonably "spot free finish" using the system. One that is waxed quite a bit and kept brand-new so it beads up a lot might not...and the effect using other soaps (or no soap?) will weaken the "no spot" effect. Even so, the filtered-water is much better than the regular hard tap water for me.