What is the difference between ${{F}_{g}}=mg$ and Universal Gravitation?

Hint: To answer our question, we will first need to understand what is meant by force of gravity and Universal Gravitation. Basically, force of gravity is the term used to describe the force by which Earth pulls an object towards its center and Universal Gravitation is the force of attraction between any two objects in the universe. We shall use this definition to answer our question in detail.

Complete step-by-step solution:
 Now, the first term in our problem is mathematically stated as:
$\Rightarrow {{F}_{g}}=mg$
The left-hand side of the above equation, that is, ‘${{F}_{g}}$’ is a term which is widely used in place of the Force of gravity due to Earth.
Now, let us suppose that the mass of the Earth is ‘M’ and its radius is ‘R’. Also, let an object be placed at the surface of the Earth having a mass of ‘m’. Then, the force by which the object is attracted towards the center of the Earth is given by Force of gravity and its value is:
$\Rightarrow {{F}_{g}}=\dfrac{GMm}{{{R}^{2}}}$
Where, G is the universal gravitational constant.
Now, when this force is equated to ‘mg’, we get the value of the term ‘g’, gravitational acceleration due to Earth at its surface. Hence, the force of gravity is a specialized term used to mostly describe the force felt by an object due to Earth’s gravitational pull.
Now, the second term in our equation is ‘Universal Gravitation’. This term is used to define the force of attraction between any two objects due to their gravitational pull, placed anywhere in the universe, irrespective of their mass and distance. It is a more generalized term that can be applied on any two bodies.
Thus, from the above statements we can see the difference between the two notations and conclude that the force of gravity is basically a subset of Universal Gravitation.

Note: It should be noted that we were asked to point out the difference between the force of gravity and Universal Gravitation and not Universal Gravitational Constant. These two terms are totally different from each other, one being a quantity of force and one being a proportionality constant. So, one should be very careful while answering problems like these.