He doesn't ride a bike whilst wearing a helmet.
Means that either he doesn't ride a bike at all, or that whenever he does ride a bike, he doesn't wear a helmet.
He rides a bike without wearing a helmet.
Means that he does ride a bike, but whenever he does so, he doesn't wear a helmet.
He doesn't cross the road whilst talking on a mobile ('phone).
Means that either he doesn't cross the road at all, or that whenever he does cross the road, he doesn't talk on a mobile.
He crosses the road whilst not talking on a mobile ('phone).
Means that he crosses the road only so long as he's not talking on a mobile.
As you can see, if you say "He doesn't ride a bike whilst wearing a helmet" it's somewhat ambiguous as to whether you're referring to the whole action of "riding a bike whilst wearing a helmet" or merely to the second part. If you want to make it clear that he might sometimes ride a bike, but never with a helmet on, you'd be better using the second form - "He rides a bike without a helmet" - or changing the order around, "He doesn't wear a helmet whilst riding a bike."