The advantages of the Pinion are that the weight and complexity are in the middle of the bike and not being bounced around in the hub of the rear wheel, especially for a bike with suspension (the unsprung weight is lower). Also, because it sits in a custom housing it can be whatever size and shape they want, rather than being constrained by the width of the rear hub and needing to be both circular and balanced.
As David says, it's easier to switch the rear wheel out either when you are fixing a puncture or because you have several rear wheels (different tyres or whatever). You could take this to the extreme with a fat tyre bike and have a skinny standard 700c rear wheel for around town and a 4" fat wheel for off road. I am considering this for my recumbent touring tandem, as the extra weight of a pair of spare wheels might be compensated for by the ability to switch from skinny road tyres to proper offroad tyres and wheels.
One more subtle advantage of the Pinion is that ground speed doesn't matter. Rohloff hubs become noticeably less efficient at high speed, but one of the places they're most desirable is in velomobiles and other speedy bikes. This is compounded by those bikes often using smaller rear wheels (406 is common, 355 is about as common as 622) which further increases the rotation speed of the hub. With a Pinion that doesn't matter, the gearbox always rotates at your pedalling speed.
The advantage of the Rohloff is that it's proven technology, parts are fairly widely available and the wheel will fit into a standard bike frame. Especially for long distance touring, the thought of being stuck somewhere remote with a broken Pinion is horrifying, because the only option is wait for a new gearbox or bike to be shipped to you, or give up and go home. The Rohloff alternative is both less likely (because we have seen Rohloff hubs do 100Mm trips quite often already), and less terrible (because you can convert back to derailleur gears or even single speed using standard bike parts). I'll hit 100Mm on my Rohloff in sometime the next couple of years, it's one of the early-ish ones (4 digit serial) and it's been trouble-free despite regular use in load bikes as well as touring bikes.
The bottom bracket gearbox has to deal with much greater torque than the hub does. Rohloff require a 2.5x gearing down between the cranks and the hub, so all the parts can be correspondingly smaller and lighter. A fit cyclist can put a surprising amount of torque into a bottom bracket, akin to a decent car engine (but at much lower rpm). Schlumpf had a problem at one stage with their sintered gears shattering in use (I experienced that), largely because they underestimated just how much force was involved. This makes the Pinion more complex to design and build for a given weight target, and probably more expensive as a result. That it can be bigger and irregularly shaped counters this to some degree.
Finally, the Pinion really can't be sold as an after-market part because you need a framebuilder to cut your bike up and fit the gearbox. It's quite possible to fit a Rohloff to almost any bike whenever you have the money available to buy one. And if the Rohloff doesn't work for you, you can take it out and sell it second hand.