Why Specific Latent Heat Of Vaporization Of Substance Is Greater Than Its Fusion ? Thermal Energy Required To Raise Temperature From Melting Point To Boiling Point. Or Increase In Potential Energy Of The Molecules Is Greater On Boiling Than On Melting

The answer lies in the arrangement of solids,liquids and gases(kinetic model).The arrangement of particles in a liquid is mostly the same as that of a solid, the only difference being it's not in a specific order. For a liquid to become a gas, there is a considerable change undergone concerning the arrangement and spacing. Hence the specific latent heat of vapourisation is higher than the specific latent heat of fusion.
Hope that my answer has been of any help.

The distance between molecules between liquid and solid states is not significantly different. In the case of water the solid ice takes up more space than the liquid water at the same temperature. All that is happening is a structural break down when the solid becomes a liquid. The difference is order and chaos. On the other hand the distances between molecules in liquid and gaseous state changes by a huge amount, so that the material expands many times when converted to gas. So in a pressure cooker the water becomes unable to expand into steam and the temperature rises above the normal boiling point of water, yet water remains in a liquid state because the expansion needed to form more steam is not available. Something has got to give and if the pressure valve goes a huge volume of steam is released as the water rapidly boils away. Gaseous material has overcome the bonds of attraction between each molecule and put a greater distance between the molecules. When condensation occurs the molecules come closer and the attraction accelerates each molecule as it comes together providing a form of kinetic energy that we call heat. This increase in the speed of the molecules is greater because the acceleration takes place over a greater distance when a liquid is formed from a gas than when a solid is formed from liquid.

I am totally unqualified to give this answer - so I hope somebody else will come in and give a better more scientific explanation

This is because of the increase in potential energy that  is larger in boiling point than in melting point.