Basically, it sounds warmer and is easier to get to the higher frets, and has more sustain. Another "benefit" (this can be debated) is that the material the neck is made of will influence the sound the most as it is closest to the pickups.

edit: They are also extremely stable in construction, but if your neck gets screwed up it may be basically impossible to correct as it is essentially the core of the body.
Yup. somewhat warmer tone. depends on neck material though. less of a heel where the neck meets the body so easier access to higher frets and it has better sustain because your neck piece and the body have more surface area to transfer the resonance. That said after owning a neck-thru i went back to bolt-on. just felt better to me. probably because i was used to it? i dunno. just a personal preference thing.
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They have more sustain as far as I know. In an interview Glen Benton said they go out of tune more easily than bolt-on instruments.
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no more sustain- that's glue in there.

glue is a bit of a muffler.

bolt ons, made properly, sustain the best.

you get smoother sound and cool looks. and the neck heel is usually better.

yup. thats it.
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I can see how the glue would affect it, but wouldn't the sustain also be helped by the fact that the nut, tuners, and bridge are all attached to the same piece of wood, with no joints in between? Doesn't seem like there'd be that big of a difference between the two, honestly.
Nope, no sig here.
A neck-through the body design is supposed to give you the advantages of a set neck over a bolt-on neck (if there are any), but without the weaknesses of a set neck joint. Since the neck and the body are basically one piece, there is no stress on a glued-in neck joint that could one day lead to problems. Whether or not there is a noticeable difference in sound between the three different methods of construction is debatable. All three methods work just fine. A neck-through design tends to be the most expensive of the three because it is the most labor-intensive to build.
There's pretty much no evidence to suggest sustain and tuning stability is positively affected on neck-through basses. Neck through necks obviously cannot be replaced either, but if broken above the heel area, a neck through can be converted into a bolt-on or set neck if the break is right, and the luthier experienced enough.