What makes sound and heat distinct ?
What makes them distinct is that sound is the transport of mechanical energy from one place to another in the form of mechanical longitudinal waves, whereas heat is defined as the transfer of energy from one substance to another due to temperature difference. Further explanation follows.
Sound is a mechanical wave associated with the vibration of some medium (some form of matter (solid, liquid or gas)). It is a longitudinal wave in which the displacement of the medium is parallel to the propagation of the wave. Google up longitudinal waves and check out the explanation of their association with sound on the Hyperphysics website.
Heat is energy transfer between substances due solely to a temperature difference between the substances. It is not the "jiggling of molecules" of a substance. That jiggling is the translational, rotational, and/or vibrational kinetic energy of the molecules of the substance and is considered to be part of the internal energy of the substance (The kinetic energy component. There is also a potential energy component).
The confusion regarding heat is that although it is not the jiggling of the molecules themselves, it is the mechanism for causing the kinetic energy of the jiggling of the molecules of a higher temperature body (where the jiggling is faster) to transfer to a lower temperature body (where the jiggling is slower). This transfer can occur by conduction, convection, and/or radiation. The first two require a medium (solid, liquid or gas) where the molecules of the hot and cold bodies interact with one another so that the higher kinetic energy molecules gives up some of their kinetic energy to the lower kinetic energy molecules. Heat transfer by radiation does not because it is in the form of electromagnetic radiation (transverse waves) capable of traveling through a vacuum with no physical interaction between the hot and cold bodies.
Hope this helps.