I see conductivity in electrical terms being the transmission of energy through current and voltage. I envision thermal conductivity in terms of the exchange of energy as a result of kinetic movement of molecules and atoms.
Conductivity of electrical energy is a result of the movement of electrons (or holes) under the influence of pressure, also known as voltage.
The phenomenon of thermal conductivity is caused by the transfer of energie under temperature or pressure.
The most efficient conductors of electricity or heat depend on the movement of electrons that are free. Metals possess the highest amount of free electrons, the most effective conductors of electrical and thermal energy are made of metals.