How does a speaker blow out?

       What Erik said (regarding clipping), but: after a x-over passes that high freq distortion to the tweeter and burns it (typically: open), the energy that no longer has a path that way, is routed to the next highest freq driver, in the circuit.

                  In the case of a two-way: that would be the woofer.

        I lost count of the number of two-way systems in which I found both drivers toasted by an under-powered amp, while in the reconing biz.    Blown x-over caps, as well.

        Though a clipped signal isn't DC; it certainly can do as much damage.

         On the other hand: hearing a, "...most painful noise, coming from one of the woofers", might indicate a woofer that was driven past it's mechanical excursion limit (X_{mech, per Thiele-Small)}, bottoming the voice coil, or: causing it to hit the gap's top edge(s).

         That type of damage usually resulted from boosted/heavy Bass and a vented cabinet, regardless of amplifier rating.

        Gotta love those pedal notes!

        Especially, on a Direct To Disc recording.

       After having repaired a few hundred speaker systems; I can only speak from my own experience/observations.

       A clipped signal and it's energy, after opening up a tweeter and having no where else to go, will end up in the x-over's, next-highest freq, driver.

 More likely to burn the first coil in the low-pass filter than the woofer itself, but things happen. :)

       Lots of blown up electrolytics, BUT- in over 15 years of repairing speaker systems (mostly: college student or Pro musician owned, it always seemed): I never once saw a burnt inductor.    Some: with signs of having been abused and accompanied (in series) by a burnt driver voice coil, but- never one that opened or shorted.   

       Perhaps the customers that clipped their amps, were just lucky in that regard.

       Then too: there are a multitude of systems out there, with no inductors in series w/their woofers, to block such damaging, high freq, energy.

                                              YEP (things happen)!

                                                   Happy listening!

@mikedc -

                               A test* (if you haven't already):

      *IF your problem/distortion only arises in one channel/woofer: swap your speaker systems to see if it follows the driver.                                   

                                 If not: it's probably the amp.

@mikedc -

       When you heard that painful woofer sound, during the organ music, was your source vinyl?

        Asking because: subsonics, caused by stylus travel (ie: record warp, arm/cart resonance, etc) will often cause woofer over-travel and damage, in a vented system.

                                       How old is your amp?

       Asking because: power supply filter caps (if one has gone South/leaked electrolyte) can cause noises that simulate a woofer malfunction*, at anything above low volume.      

                            *Especially on heavy Bass notes.