We've had a Chariot CX1 for 8 years (gulp), and I've also used a few older Burley versions (without suspension) for carrying two at once.
The Burleys I've used were great, but were seriously outshone by the Chariot in terms of versatility and weather protection. One of the joys of the Chariot is the ease of converting it to a jogger and being able to properly run with it, on and off road. With the appropriate add-ons (waterproof cover and sleeping bag thingy) it's has regularly coped with wet, sub-zero, off-road rides, but also can be stripped back to make a cool/shady summer vehicle. Big tyres + suspension means it handles rough track really well, and I really have forgotten it's there a couple of times, with it ending up upside down as a result! Occupant unruffled - and unwoken on one occasion.
That said the latest Burley looks very similar in spec now. The key difference seem to be the seating position and weight distribution. The child sits a bit higher and more upright in the Burley (good for seeing out), whereas they've lower to the ground and more reclined in the Chariot (better for handling and good for sleeping) - hence why the side windows go lower on the Chariot. The Chariot also gives more leg room for running. I like the Chariot hitch better than the Burley hitch too.
I suspect that price and local availability/support are the key factors to consider. When we were buying the Chariots were pricier than the Burleys (our CX1 came to about 800 with all the bits), but now the specs are closer I suspect the prices are too.