The main advantage of neck-through construction is better sustain, achieved through greater stiffness. It's all about maintaining the string's energy as long as possible.
Why does a guitar string lose its sustain? Why doesn't it keep vibrating forever? When you pluck a string, you impart energy to the string, and that energy keeps it vibrating. But some of the energy is transmitted through the bridge and the nut/fret into the guitar, and as that happens, the string loses its energy and vibrates less and less until, for all practical purposes, it stops vibrating entirely.
So how to you keep a string vibrating? You want the guitar to absorb less energy into itself and instead reflect more of the string's energy back into the string itself. A guitar with more mass and stiffness will reflect more energy back into the string than a lighter-weight, less stiff guitar. This is why, say, Modulus Graphite instruments have such sustain: their carbon-fiber necks have way more stiffness than wooden necks---so much so that they don't even require truss rods.
A bolt-on is inherently less stiff than a neck-through because the neck joint can't have as much stiffness as the two pieces being joined (the neck and the body). Whereas a neck-through is one solid piece of wood from nut to bridge, so it's stiffer and thus reflects more energy back into the string.