A J:
A P:
The jazz sounds OK, but that P is exactly what I was talking about. That thing sounds like rotten butthole.
As an example of what a Warwick is capable of, this is Nick Schendzielos of Cephalic Carnage, Havok, and A Job For a Cowboy.
Here's Rob playing Orion on a Warwick (not his P)
Here's Erlend Caspersen of Blood Red Throne, Emeth, and Deeds of Flesh rocking a Warwick in a way I have never seen or heard on a Fender of any kind.
Here's Alex Webster with a Spector:
If the difference is in the attack. If you are playing quick you need a tone that articulates and a neck that keep up with you. A P is a solid tone in the low and mid register but stinks at the high, has lousy access, and a neck not known for speed. P basses articulate in a different way. A Jazz bass articulates well, which is why they are preferred for slap, but are limited in their mids, and still lacks 24 frets or a comfortable heel, if that's what the OP is after.
The OP never asked "What's the best bass for metal". He had gave some decent specifics on what he was looking for and he already said it's unlikely he can get the same type of Aria Bass Burton played. I'm sure it would be different if someone said they wanted to go for a Victor Wooten or Anthony Jackson sound but couldn't afford a Fodera.