Isn't this like comparing bath towel to a laptop? They're completely different animals.
You can basically just become enamored with both a jazz bass and a bongo bass and find yourself wondering which one to actually buy. It doesn't particularly matter that they're very different. You still need to choose. So it can be helpful to have some people with experience spell out the differences, especially the less obvious ones that might only reveal themselves in live playing.
Basically I would say go with the Jazz bass. I've owned and gigged both quite a bit. The Jazz bass has the natural, inherent Fender midrange that makes it always work. It'll stay in the pocket but you'll be able to hear it and it'll punch without having to literally boost the midrange using EQ. It'll work equally well with fingers or pick with no knob turning needed. The Bongo is not quite so easy. I'll paraphrase some thoughts I've posted in the past. I would describe the Bongo as having a fundamentally weak sound that relies heavily on the preamp to cover for it. With a Stingray it's the perfect marriage of preamp and pickup. Each needs the other and they contribute equally. With the Bongo there is a sense that it's nearly all preamp with not much substance behind it. The pickups are not positioned such that they are viable when soloed. So you always need to be working with a blend of the two. I always found that the Bongo was either a bit too woofy and EB0-like, or a bit too Jaco. It was tough to find the optimal sweet spot. I believe I ended up settling on favoring the neck pickup about 75%, simply because anything else was either too scooped or too nasal and thin.
I have theorized that part of the issue I have with the Bongo comes from the high headroom of the preamp. You can sort of saturate a Stingray by playing hard, but that doesn't happen with the Bongo. It's the feeling of needing to turn up the gain on your amp, or raise the pickups closer to the strings. I could never get it to deliver the rich, intense digging in sound that I wanted, and had to simply compensate by turning up the preamp EQ. That's my big critique of the Bongo, it always felt and sounded like the pickups were too far away from the strings, even though they weren't.
I'll also say that the midrange on the Bongo is odd. It's one of the few basses where I reliably dislike the sound when I hear it on YouTube and elsewhere. It's like the midrange sounds like it's being pushed, but the effects of scooping the mids remain. For example, playing with a pick doesn't achieve the same volume as playing with fingers on the Bongo generally. Or it takes on a more underlying type of presence in the mix than you would expect given what sounds like adequate midrange. The only way to get it out of that would be to boost the mids to an exaggerated degree or favor one pickup, in which case you quickly start to run into the too woofy/too Jaco dilemma.
I'm not sure I'd be persuaded that the Bongo's B string is that much better than the Fender's would be. My Bongo 5 had the same exact B string problems that every 34" B string has. Namely that it needs almost a completely different setup just for that one string to make it ring out, sustain, and sound as rich, powerful and present as the other strings. I think that's pretty inescapable at 34". It can be made to work fairly well but it's not somehow magically immune to physics. And don't forget, with the Bongo 5 comes the slipping off the fretboard G string thing. Yep, that's in the Bongo too.
Obviously this sounds fairly harsh toward the Bongo, but it's a bass I gigged for about 50 shows. I'm fairly familiar with it and have no bias. I think it's one of the coolest looking basses ever. With a different preamp and pickups it would probably transform entirely. Also, if it were a single H Bongo, that's a different story entirely as that one pickup is in the Stingray spot. That's a much stronger position than the ones chosen for the HH.