Overall, the quad is a bit safer. The kids' quads these days are pretty well built, and have some decent suspension. The baby Predator we got has double the suspension travel of the others and is quite stable (it can still wheelie with a grown 'man' on it, though heehee).
We initially got an XR70 for our daughter (my step-daughter). She was having a bit of a hard time with it, and was getting discouraged everytime she fell over and got a bump (girls these days

). Her biological father suggested we trade it for the quad, and he'd pay half the difference, so we did. She is much happier with it and while there is still the chance of injury, she has yet to experience the joy of falling from one's quad.
My personal take is kinda torn. If a kid starts out learning to balance on a bike, they can have a lot more fun as they grow with it. But, it also comes with a lot more broken bones.
With a four wheeler they can still get out and have fun with the family with less overall risk. Sure, you can have the freak accident where they flip the machine over and it lands on them, but the chances of injury on the four wheeler are much lower than with the bike.
Just my observation.