What is the differentiating factor between work and heat?' Although work and heat do the same thing (increase or decrease the internal energy of the system), There is still a fundamental difference between them. For example, The way in which entropy is defined is a very good way to differentiate between work and heat. But, why is there such a distinction between the two things? Is it the limitation of Newtonian mechanics that it never accounted for something like heat which could also change the energy of the system? Is the word "Thermodynamical work" or "Hidden work" suitable for heat?
 A: This is a very good and fundamental question. It lies at the heart of thermodynamics and related disciplines. There is no general answer for all physical systems, unfortunately.
What we define as work and heat is highly dependent on the system in question. Take the classical example of gas confined by a movable piston, in contact with a heat reservoir. In this case, all energy transferred to/from the gas from the mechanical movement of the piston is work. All energy transferred to/from the heat reservoir is heat. We do not prove that the system only exchanges heat with the reservoir; we define it in this way.
The take-away point here is that we define heat and work at the outset, when we are developing our physical model. Where the concept of heat is useful is in keeping track of changes that result in changes in entropy, so we usually associate heat with entropy. Usually, the definition is of the form 'heat is energy exchanged with the heat reservoir', so we need to have some notion of a heat reservoir in our model, for which a temperature can be defined. Then:
$$
\partial S = \frac{\partial Q}{T}
$$
Unfortunately many common definitions of heat that are given in various sources are circular in nature; for example "heat is the energy that is transferred due to temperature differences", or "heat is the energy that can't be used to do useful work."
A: Consider a gas inside a container. Every time a molecule hits the wall of the container there is work being done. If we were able to compute the force it did and the displacement it caused to the wall and then sum over all collisions we would have a macroscopic value for work and we would not need to talk about heat. The first law would be just $\Delta U+W=0$. Since we are not able to compute all those microscopic work we effectively call the missing quantity, $\Delta U+W\neq 0$, heat. Heat work done at molecular level.
A: 
What is the differentiating factor between work and heat?'

Both heat and work are means of energy transfer. The main differentiating factor is heat is energy transfer driven solely by temperature difference whereas work is energy transfer driven by force acting through a distance.

For example, The way in which entropy is defined is a very good way to
differentiate between work and heat. But, why is there such a
distinction between the two things?

Entropy is only transferred when there is a reversible transfer of heat. However entropy can be generated by irreversible work or by an irreversible transfer of heat. So entropy is not unique to heat transfer.

Is it the limitation of Newtonian mechanics that it never accounted
for something like heat which could also change the energy of the
system?

Newtonian mechanics accounts for macroscopic changes in the energy of a system relative to an external (to the system) frame of reference, such as changes to the kinetic energy and gravitational potential energy of an object as a whole. Heating such an object would not change its velocity (kinetic energy) or its elevation (gravitational potential energy).
Thermodynamics, in contrast to Newtonian mechanics, accounts for changes in the internal energy of an object, i.e., changes in the kinetic and potential energy of the object at the microscopic (molecular) level, which may occur as a result of work or heat transfer.

Is the word "Thermodynamical work" or "Hidden work" suitable for heat?

I wouldn't say that "thermodynamical work" or "hidden work" are suitable terms for heat. But it is true that when one heats an object or does thermodynamics work on an object one can not "see" the molecules of the object speed up or slow down (change kinetic energy) or see the molecules move closer or farther apart (change potential energy) as a result of the energy transfer. On the other hand, when one does Newtonian work on an object we can see its velocity change (change in kinetic energy), or see its elevation change (change in gravitational potential energy). So, in that sense, the effects of heat and thermodynamics work might be considered "hidden".
Hope this helps.
A: It turns out that terms representing the kinds of energy in a system generally are products of two factors.  One factor is an extensive variable (depends on how much of the system you consider), and the other is intensive (does not).  For example, $PV$: $P$ is intensive, $V$ extensive, or  $\mu N$: $\mu$ (the chemical potential of a species) is intensive, $N$ (the number of molecules of the species) is extensive, or $VQ$: $V$ (not volume, but electrical potential) is intensive, $Q$ (charge) is extensive.  These pairs are called "thermodynamically conjugate."
An infinitesimal change in energy of the system can occur either with a change of the extensive variable or the intensive one:  e.g. $-PdV$ or $-(dP)V$.  For any extensive variable other than entropy, when the extensive variable changes, work is being done (e.g. $-PdV, \mu dN, V dQ$). This is analogous to (and sometimes equal to) a force acting over some distance. Any additional change of internal energy (e.g. $dU + PdV - \mu dN - VdQ -$ (other changes associated with extensive variables)) is associated with heating, and equals $TdS$.
A: Heat is a form of energy. And if we go into the basic definition of energy, It is the capacity to do work. They are interrelated. But mathematically they are one and the same. Their SI units is the same ie. Joules . The only thing that changes is the physical interpretation .
Think of a heat engine in a car. It is ready to do work because it contains a certain amount of stored energy . When work is done, energy is transferred between systems or from one form of energy to another.
Thus Heat energy is what the engine contains ( or possesses) while work is what has happened with that heat energy , is it lost to the surrounding, or is it converted to kinetic energy? If yes, how much is what is Work  . You will be knowing the formulas that work is the difference in potential energy or sometimes difference in kinetic energy.
So finally, Heat is what you already have and Work is what you do with it
