Regarding this P vs J conversation (or any other kind for that matter), it depends on the player and their combined musical sensibilities, experience and execution of musical ideas. That is obvious. But the discussion about instrument choice that is often perceived as frivolous is an important one nonetheless. An instrument undeniably has a big impact on how we feel about the music. It can also ultimately affect our employability. Thank goodness we are all very sensitive to these things--which is one of the many reasons we have chosen music as either a hobby or profession and also find ourselves on this site discussing our tools. Note the distinction: we can have a conversation about our tools, or we can have a conversation about our musical choices; sometimes those conversations intersect, sometimes they don't. One shouldn't lose sight of the fact that instruments are tools and are easy to account for with words. Truthful and inspired music making cannot easily be conveyed with words. So I am opposed to calling the P vs J conversation frivolous; it's just a small part of a larger conversation that justifies its own thread.
I can only offer thoughts on this subject pertaining to my direct experience, as we all can, naturally. I am a typical professional bassist who is originally self-taught, went to music school, studied jazz, classical double bass and who has played in all kinds of pop, musical theatre, rock, jazz, church, whatever. Mostly I'm a tradesman, just as a car mechanic or plumber is. I gots tools and experience and skills, and I got to pay the bills. And buy some new toys once in a while.
Getting to the point, I own both P's and J's. I mix it up all the time! I like lists, so here are the questions and considerations I ask myself before taking one or the other to the gig, and keep in mind I a choose to be more of a utilitarian player than an original artist:
What is the music at hand?
How was it recorded in the first place?
When I practice the tunes at home, which instrument sounds most appropriate?
Does my instrument choice serve the music well?
Who are the the other musicians on the gig?
Are they even sensitive enough to hear a difference?
What is my function in the group? Am I musical director or purely hired support?
Do I need to be playing a bass that allows me to quickly dial in an assertive tone to cut through the mix and carry the rhythm section, or can I trust the rhythm section to carry their own weight?
How many instruments onstage?
Does the piano player or guitar player know how to stay out of or compliment the bass player's range?
How does the venue sound, and how does that affect how my bass sounds out in the house?
Will I be using an amp, going direct into the house without an amp, using in-ears, or a combination of those?
How is the sound engineer?
How is the drummer?
Does the drummer know how to tune his drums well, especially the kick drum?
Will I be playing with backing tracks, and how do they sound?
Will I need more versatility from an instrument, such as a 5-string active jazz bass, to best accomodate those backing tracks from song to song?
Is it a very long gig? (in which case I might bring a 7.5 lb P bass)
How is the weather stability? (If playing oustide in the heat or inside with freezing cold AC, perhaps I'll chose a bass with graphite reinforcement rods in the neck, or even a full-on carbon fiber neck.
There are probably many other considerations that I haven't listed, but this is part of my thought process. And I'll tell ya, the other musicians in the group usually don't want to hear about this stuff. In fact good musicians (who are gearheads but might not admit it) are wary of gear heads. So one approach is to think about all that stuff a lot, but DON'T let on that you do, and make it your business to put everyone at ease. Instrument choice is ultimately only a fraction of the larger picture when you take into account that the other musicians are typically equally involved in their own world. It's a larger part of the equation when it comes to your own perception of your musical identity.
Ideally, I'm playing with good musicians who don't get hung up which instrument I play. They trust that I'll make the right choice whatever that is, and I trust them to do the same.