Why doesn't the sundew have to move as quickly as the Venus flytrap to imprison an insect?

Sundews excrete a sticky mucilage from their leaves which ensnare insects that come into contact with it. Because the leaves are sticky enough to trap most prey, the leaves do not need to close quickly. The prey is effectively disabled before the trap closes. It is like having a glue trap to catch mice.

Venus Flytraps do not have this sticky mucilage and must snap closed fast enough to prevent the prey from escaping. This is more analogous to a traditional snapping mouse trap.