The CP will sound similar to a current Gibson Thunderbird (according to folks here that have both).
The VP will sound more like a vintage 1960's Thunderbird (according to folks here that have both).
Since I don't have a CP or a 1960's bird I can't confirm that, but assuming it's true I can offer an opinion on my VP vs my 2014 Gibson TBird that should answer your question.
The VP sounds very different from the modern Gibson TBird. It's a more agressive in your face type tone, more grit, more growl, more grind. It wants to be up front in the mix but with the right strings you can get it to lay back a bit. The Gibby on the other hand is a little more refined, smooth and laid back. It can still grind but you'll have to dig in deeper and have the right strings. To my ears, the Gibson TBird sounds like a P bass on steroids and the VP sounds like a TBird on steroids.
Physically the VP is a couple of inches longer but most of that is in the headstock (which is huge). I suspect the reach to the G string tuner is longer on the VP by an inch or so. The hardware on the VP is quite different - pups in chrome cans with trim rings and a two piece bridge vs black plastic pups and a three point bridge. The VP bridge alone might be enough to sway some people in its direction (there's kind of a love it or hate it mentality with some players around here regarding the three point).
Price wise you'll pay more for a used VP than a discontinued CP but probably only a couple hundred more. Both are great basses. I personally like the VP better than the CP and every bit as much as my much more expensive Gibson TBirds.