Here is my limited experience. I'm fairly new to bass and own two quality basses, a Fender Jazz and a Fender P. Both are wearing flats at the moment. I use them for home recording my solo material. I like the P but don't love it. I love the Jazz but don't always like it, if that makes sense...
The P is simple to record and EQ. What you hear is what you get. It fits in with most of the music I play.
The Jazz took me a while to learn to record properly. The Jazz--my Jazz anyway--has an inherently scooped tone that can sound thin. I've found that if I am not careful, the body of the sound disappears in the mix leaving behind the boomy and trebly parts.
For me, the key to getting a full sounding Jazz tone is to roll back the pickup knobs and/or tone, while boosting the lower midrange on the amp EQ. Usually I'm playing with one of the volume knobs at about 75%, and the other slightly lower or higher. The lower the volume knobs, the higher goes the tone knob and vice versa.
There's always a bit of dialing in that takes place before I track the Jazz. When I get it right, there is a magic that I just don't hear with the P. It has a warm, deep tone that still retains detail in the higher frequencies.