I own a 2000 American Standard Jazz AND Just bought the new AS2008 P.
Why does it matter for comparison? I've heard the necks on the JV and PV are the same.
IF that's the case: Here's what I can tell in terms of the neck.
The new 2008 has a slimmer neck from front to back, making it a bit more 'comfy'.
Now, I love both basses, and the neck size is not a deal breaker in any way, the 2000 is very playable, but the new 08 neck feels like 'home'.
The 2008 has smaller headstock, due to the G String Tuner on the bottom.
The new 08 Tuners ARE lighter than the old ones...and look MUCH better than the hollow 00 version, but this is cosmetic at best.
The new 08 is a HELL of alot lighter.
The 2000 Jazz will need the pickups replaced. IMHO, but there it is, I put the Nordstrand NJ5-F in, the tone smokes now, before, the tone was well, it didn't sound like a Jazz to me if that makes sense. Now it does.
Both basses can be strung from the back, but the bridges are vastly different, 00 being more 'traditional'.
Good luck