Idle fuel flow on a 737NG is around 400kg/hour/eng. Most of that (at ground idle) is used just keeping the fire going, and the metal spinning rather than producing useful thrust, so you can reduce your taxi fuel burn by about 40% by shutting down/not starting an engine.
A short, 10 minute taxi thus saves around 30 kg (round figures = £20), which doesn't sound much, but with a fleet of 100 aircraft, flying an average of 4 sectors each, 365 days per year this = nearly £3 million! Save fuel on a 20 or 30 minute taxi and you are really saving cash.
In terms of Risk Assessment - starting the second engine on the move is a crew distraction which needs to be managed with solid procedures. The fire risk is the same (i.e. negligible) and the evac/fire engine access is probably better on a taxiway than on stand - given the better access to the aircraft and clear area for the slides.
The difference between Boeing & Airbus is that the Airbus start and abnormal-start shutdown is more automated (read, less chance of damaging an engine whilst distracted) than the Boeing.