A ported box is like a beer bottle. Blow across the mouth of the beer bottle, and that wuuuuu tone is the Helmholtz resonant frequency. Notice that it doesn't die out instantly when you stop blowing. We could call this "overhang" or "ringing" (though it's a lot lower in pitch than the ringing we associate with a bell).
A ported box uses its Helmnoltz resonance to extend the bass deeper than it otherwise would have gone, but it also tends to resonate (ring, overhang) for a little bit after the cone stops moving (the cone's motion is analogous to blowing across the mouth of the beer bottle). In fact, as the energy decays, the pitch can shift in the direction of that Helmholtz resonance. So not only do we have overhang blurring the tail end of the notes, we can also have the pitch shifting as the note decays!
As far as the cone stopping, the amplifier/speaker interface plays a role there, but note that the cone is never unpowered. The stiffer air spring of a small sealed box doesn't give it better transient response than the weaker air spring of a large sealed box, in fact, the transient response of sealed boxes improves as the box gets larger (air spring gets weaker). But with a ported box, the box itself keeps on singing its one-note song, and that's the culprit.
So what kind of cabs do I build? Ported ones. I'm largely trying to get a lot of sound-per-pound and sound-per-size, and also in general I prefer the top end performance of woofers that are better suited for ported than sealed enclosures, this because of their more powerful magnets. If a ported box is sized and tuned so that it's not relying too heavily on the port's output, then the problems of overhang and pitch shift are reduced. This is especially important for critical applications like studio monitors, which I sometimes do on a custom basis... the engineer has to be able to clearly distinguish between the bass and the kick, for example.
I've had the good fortune of spending some time with a few of Mike Arnopol's manipulated vortex waveguide cabs, and of using the same technology in a different market. In my opinion conceptually they are sort of like a fuzzy-length transmission line, in that they make use of the backwave energy, but they a) go deeper than a comparable-length transmission line does and b) don't have the one-wavelength cancellation dip that we see in measurements of transmission lines. In my opinion, they have low-end extension that is competitive with a ported box but without relying a Helmholtz resonance, so they have very good note definition even at the low end of the scale. I have to concede that this is an area where his cabs outperform my ported cabs, and if I built sealed cabs, his cabs would outperform mine in low-end extension and authority. In fact the best low-end impact and definition I've heard in any of my home audio speakers has come from one that's essentially a copy of his MAS-210. That technology works well with woofers whose motors are too powerful (Qes too low) for an optimum vented box.
I mentioned transmission lines. They have a size and weight and cost penalty, and at the frequency where the line length is equal to one wavelength, the output from the line emerges 180 degrees out-of-phase with the output from the front of the cone. This causes a dip. Dips are less audible than peaks, and often go unnoticed on recorded music, but they can make certain notes die so you don't want them in a bass cab. There are techniques that minimize this dip, but in general the net result is a big cab that won't go as loud as other big cabs. But they can sound really good.
This past weekend I showed a pair of home audio speakers at the Rocky Mountain Audio Fest in Denver. The hotel rooms we use are notoriously boomy. We used a variety of techniques to deal with that. We tuned our ported boxes to different frequencies, so they would not all Helmholtz in unison. We plugged ports, converting ported boxes to sealed boxes. We even used polarity reversal to get some cancellation going on. But if the venue had been outdoors, we would have unplugged all of the ports and gotten as much low-end assistance from Mr. Helmholtz as we could. Point being, what works best is often room-dependent.
Someone mentioned using my cabs with some of the ports plugged. A vented box tuned much lower than optimum, whose internal volume is smaller than needed for a true EBS (extended bass shelf) tuning, can approach what might be called "sealed box emulation" tuning. That would be a rather benign ported box, as far as overhang goes, but still not as good in that respect as a sealed box or one of Mike's cabs.
Anyway this is a really fun topic and I've enjoyed reading this thread. Thank you Mark for bringing it up because I'm benefitting from learning what bass players like. Maybe one day I'll take a shot at a sealed box 3015LF system.