lighter bikes, within reason tend to be just as sturdy as heavier ones, because they are higher quality. now, sure, a ultra light roadrace bike wont be as strong as a clunky 80s steelframe mountain bike, but a modern high end carbon fiber mountain bike is probably stronger than that old junker.
lighter bikes accelerate faster, brake faster, turn faster, climb faster. better quality bikes will have lower friction losses from bearings, drive chain. thinner high pressure tires have less rolling resistance wind resistance.