Another way to think about the process is graphically. If we look at how the temperature changes as the ice is heated, we would see a graph that looks like the following:
A graph like this is called a heating curve. (The word curve here just means that it’s not a line, not that it’s curvy.)
Notice that the plateau (the flat part) when the ice is melting is shorter than the plateau when the water is boiling. That’s because the ΔH fus is smaller than the ΔH vap. Remember that the ΔH’s represent the change in the potential energy from one state to another, and that potential energy is about attraction and distance. Since the change in distance between molecules when going from solid to liquid is smaller than the change in distance going from liquid to gas the energy change is smaller and, therefore, the ΔH fus is smaller than the ΔH vap. So when you are heating something, it takes less heat (and therefore less time) to melt it than it does to boil it.
A cooling curve (what happens when heat is removed) can be seen here.