A brief rundown of the advantages and disadvantages of each driver's vehicle, followed by an (admittedly subjective) evaluation.
Axel's cab is the well-rounded choice. It's "generically good" in nearly all respects, but at the same time doesn't have any qualities in particular that you can maximize/abuse.
The light weight of the cab means that when you hit another vehicle, chances are higher that you'll be either knocked way off course, turned completely around, or fall victim to the dreaded teleport bug. The wide width hurts your ability to squeeze between two vehicles without bumping them; making traffic navigation harder than it could be.
These two cons aside, I can't hold very much against this cab; but there's no compelling reason to select it either. There are better choices, but this vehicle still gets a half-hearted thumbs-up.
The big draw of this cab is the fact that it has the fastest normal floored speed. Thing is, once you've learned how to Crazy Dash and Limiter Cut, this "advantage" is totally negated. That's when you start noticing the things that are wrong.
The primary problem is the combination of this cab's large size and awful handling; it's a boat! When driving at Limiter Cut speeds, weaving in and out of traffic is difficult enough; complicating matters is the fact that this vehicle is wide, and built to slide! This makes it very easy to execute Crazy Drifts (often unintentionally), but becomes a huge problem when all you need to do is quickly drive around a car and get back in the same lane but instead wind up sliding off the road or into the path of oncoming traffic (thank goodness this cab is heavy!).
In Crazy Taxi, "threading the needle" between vehicles is of the utmost importance, both to build up the value of your Tip Multiplier Bonuses and to ensure Speedy ratings to prolong your playing time. It's more difficult to do this with B.D. Joe's cab than any other; you have to read traffic well in advance so you can (hopefully) avoid tight situations instead of attempting to navigate through them. Unfortunately, this isn't always possible. I'd only recommend choosing this cab if you want to challenge/handicap yourself, or aren't shooting for the highest score possible.
By the way, if this is the only cab you ever use you're making things much harder than they need to be: unfortunately, for most players this one gets a thumbs-down!
Gena's vehicle has the best acceleration and deceleration of all the available cabs. This becomes a double-edged sword when it comes to bursts of speed from the Crazy Dash and Limiter Cut. Because her cab has the fastest deceleration, you have to Limiter Cut more often than normal to avoid losing the speed you've gained. The flip side is that her cab stops more quickly than any of the others.
Another double-edged sword is the way her vehicle handles. It's extremely responsive to slight turns of the wheel due to amazingly good traction. Until you get used to it, the steering may seem overly "touchy." Stick with it, and you'll ultimately be rewarded.
This same great traction spawns another negative consequence: it is much harder for her to do Crazy Drifts. You'll often find the cab simultaneously slowing and turning instead (especially when traveling at normal floored speed). The workaround is to execute the Granny's Crazy Drift variation when you're not traveling at Crazy Dash speed or higher. Another thing to keep in mind is that the traction prevents her cab from Crazy Drifting as far as the others would.
The only negative of her cab you can't overcome with driving skill is it's light weight. This makes her susceptible to getting turned around or becoming a victim of the teleport bug after a high speed collision. You'll just have to do your best to avoid such collisions by taking advantage of this cab's manuverability.
Skilled players capable of constant Limiter Cuts can achieve extremely high scores with Gena's cab (perhaps the highest scores possible). Just take care to avoid vehicle collisions (especially the oncoming variety) and concentrate on always gaining playing time rather than running up the Tip Multiplier Bonuses with lots of Crazy Drifts.
Gena's cab is also great for unskilled players who don't know any of the special techniques. They'll get better scores than they might with the others due to its excellent acceleration (which sort of compensates for the lack of the Crazy Dash) and braking. This one gets a hearty thumbs-up for beginning and expert players alike!
Gus's cab is the heaviest one of the bunch, with the slowest acceleration and deceleration. It sounds like a pretty boring ride, but that couldn't be further from the truth. The slow acceleration actually becomes a non-issue once you've learned how to Crazy Dash. The slow deceleration also sounds bad; until you realize it means speed bursts from Crazy Dashes and Limiter Cuts last longer (so you don't have to do them as frequently).
The handling in Gena's cab is tighter and has better range, but Gus's feels more natural. This combined with his cab's thin width really helps when navigating tight traffic. A skilled driver can really run up the Tip Multiplier Bonuses on lengthy trips.
Due to this cab's heavy weight, once you're driving at Limiter Cut speeds you can afford to be a bit more daring than you can in the others. While you still want to avoid bumping into large vehicles for both time and Tip Multiplier Bonus reasons, if you must hit another car at high speed chances are good that it will be jarred from its course more than you are.
The big problem with Gus's cab is stopping it. When dropping customers off, you can usually Crazy Drift into a wall, or plow into a less solid object (hot dog cart, phone booth, etc.) while Crazy Stopping. Picking up customers is a different story; and the only answer is experience. With all of the cabs, it's better to error on the side of coming up short of, rather than overshooting, a customer. You always have to wary of this when driving Gus's cab; it doesn't like giving up momentum.
With its slow acceleration, this cab probably isn't the best for players who don't know any of the techniques. Once you can do the Crazy Dash things get better, but this one doesn't really shine until you become proficient at Limiter Cuts. (In fact, the slow deceleration may lend players trying to learn the Limiter Cut the extra time they need to separate the movements and get the rhythm right.) With just one major downside (that can be overcome with experience), this cab also earns itself a thumbs-up for advanced and expert players.