I've been struggling with this for years. I love the way a Jazz sounds individually, but often I feel that the sound gets homogenized in a band mix. That has led me to mostly favor P-basses and some non-Fender designs in recent years, even though a Jazz Bass always feels like home.
This is my EXACT issue with jazz basses. I LOVE the way a jazz sounds playing in the bedroom. The tone is just straight nasty, growly, and fantastic. Even my favorite players have used Jazz basses and their tone is money (Geddy, Timmy C). However, once I step into a live setting, I just don't get the same punch and response that I do using a P or a PJ. I've also noticed watching OTHER bands, that a player using a jazz is just not as upfront and prevalent as someone using a P or even a Stingray. It really does create a dilemma! Also to note, I say all this as a player in hard rock/metal bands.