I did some basic recording of the '66 vs a newish Highway One (with Greasebucket circuit) and sound-wise there wasn't a great deal of difference even with slightly different strings on. All nickel rounds for the '66 compared to nickel-wound rounds for the HW One. Not done the full P bass on flats thing yet.
In terms of general feel the vintage bass is lighter, has a nicer finish to the neck that tbf is probably due to it having been played over the years and so has worn down to a lovely silky finish. The profile is also slightly fatter and this suits me a bit more. When I bought it I went looking for a well used item on the basis that the fact it had been pre-loved and played as it was a good one. I think this is the case for the bass. I do gig it around the place, as much as when I had a more modern US Precision. The modern bass has long gone so I guess that re-enforces the above.
I have a couple of old violins too and they also compare very favourable to the more modern instrument I have.
It's entirely possible that there is a element of rose-tinted specs about the whole debate and the fact that these basses are costly may subconsciously mean some kind of justification filter is kicking in; who knows. I love mine and love playing it.
Desperately coveting @Burns-bass 's '66J . That would be a lovely soul mate for my P bass.