This corresponds to stronger barrier penetration. If the energy barrier is very high, then the exponential decay of the wave function is very rapid. In this case the wave function goes nearly to zero at the boundary between the allowed and forbidden regions. This is why we specify the wave function to be zero at the walls for the particle in the box. These walls act in effect as infinitely high potential barriers.
Barrier penetration is important in a number of natural phenomena.
Certain types of radioactive decay and the fissioning of heavy nuclei are governed by this process. In addition, the field effect transistors used in most computer microchips control the flow of electrons by electronically altering the strength of a potential energy barrier.