I can only add what I've learned during my Les Paul build, specifically the 1950's through 1960. The wood really does make a difference. Funny to think how the acoustics of wood relate to a signal that is generated by the pickups, but it does. As Gator mentioned, wood is a big factor in how an instrument will sound, as is the case for the Stradivarius. Turns out the wood he was using happened to be really dense, the climate had been such to promote "good wood", he didn't know this at the time though. The Honduras Mahogany used for the vintage LP's were, as mentioned old growth. Very dense stuff. While you can still buy old growth today, it's rare, very expensive and usually hoarded. Same goes for the Brazilian Rosewood that was used for the fretboards. The 2nd neck in my build is true old growth, you make sure to make careful cuts and make the most out of the wood, even scraps that may usually end up in the trash. The Honduras Mahogany on the market today is the same species, but usually grown in Asia.
Old growth V. new.... Not only does climate have an effect on the density of the wood, but the old growth stuff had huge, dense canopies. Light was not able to penetrate deep to the forest floors, so the trees grew very slowly. Today, the climate is different and the canopies are much lighter, allowing light down to the floor and letting the trees grow fast. If you can find all the old materials, you can make a guitar today, sound like a true vintage. In fact, there are stories of replica LP's being sold as the real thing, fooling many people at first, only to be found out later that it is a fake, disappointing the owner.
I think one reason the '59 Les Paul became a collectible item is do to the change in finish. All the previous LP's except 2(?) before that, had been solid colors and most hiding the maple tops - which were plain at that point. A lot of the LP's previous sound really good too. Gibson was on to something with the use of two main woods. Another famous trait of the LP is it's sustain, which can be attributed to the neck glued to the body, rather than Fender's bolt on style. There were maybe 1,700 LP's finished to the Sunburst specs from about mid '58 to mid '60 and a lot of those were destroyed and some still are hiding in the backs of closets. The reason the 58/9's are even more sought after has to do with the dyes/tints that were used and faded. This made each 'Burst completely unique.
Looking at sound/tone, there is something magical about them. But I don't think age has too much to do with it. The albums that are suggested listening to understand the tone, were all made about 10 years after the guitars left the factory. As was mentioned, the pickups were not wound as efficiently as they are today (when mass produced), so they also added character to each LP as well.
Also mentioned were how the magnets in the pickups degrade. This can happen to any guitar at any time. Leaning an axe against an amp will do it. Playing the guitar itself, causes the magnets to degrade. Any disturbance in its field. Magnets also degrade quickly at first, then at some point, they really slow down.
Small correction... "Power" isn't really running through a guitar. The strings, being steel, disrupt the magnetic field in the pickup, this field generates a signal that passes through the cable and into the amp. The amp takes the signal, boosts it and passes it to the speaker. The speaker, vibrates back and forth, creating the sound waves that turn into music.
The player and amp also contributes to the sound.
Great stuff. Listen to some of the other vintage guitars as well.
YouTube - 1959 Gibson Les Paul 2 00976
P.S.: The '58-'60 Les Paul's are rare, so the value has really gone up. They also happen to sound really good and are getting better with age. The fact that they were hand made also affects the value. The type of craftsmanship doesn't exist these days and there would be no way to mass produce them by hand anyway. Each LP has a slightly different shape to the top/arch. Based on who did the top, what the condition of the sandpaper was and pressure applied to the slack belt. Even if you could apply the same pressure to the sanding belt each time, the tops would vary as the grit wore down. Today's LP's are carved via duplicators with almost no variance to the shape, only the wood properties.