Why do we need the concept of Gravitational and Electric Potential? I understand that we need potential energy for the concept of energy conservation. However, why would we come up with a definition like 'energy required per unit mass/charge to bring the mass/charge from point A to B. The part says 'per unit mass/charge' allegedly to avoid mass/charge dependence as the potential energy depends on the mass/charge. Why do we need to get rid of the mass/charge dependence and invent a new concept like 'potential' out of potential energy?
 A: 
However, why would we come up with a definition like 'energy required
per unit mass/charge to bring the mass/charge from point A to B.

First of all, the concept of gravitational or electrical potential is that it is an absolute quantity requiring the assignment of some point a value of zero potential. It is has no real physical significance because its value depends on the arbitrary selection of a point as being zero potential.  Griffiths, in his book "Introduction to Electromagnetism",  makes the following statement:
"Evidently potential as such carries no real physical significance, for at any given point we can adjust its value at will by a suitable relocation of 0"
His statement applies equally to gravitational and electrical potential.
What really matters is potential difference, which is independent of the point where a potential of zero is assigned. The electrical potential difference, or voltage $V$, between two points is defined as the work per unit charge (Coulomb) to move the charge between the two points, which will be the same regardless of the point assigned a potential of zero. The same applies to gravitational potential difference, though it is seldom used.
The electrical potential difference, $V$, is an essential concept in electrical circuit analysis. It  gives us the electrical potential energy gained or lost per unit charge in moving the charge between two points so that we can apply Kirchhoff's voltage law for an electrical circuit. Although the electrical potential at a given point may be different depending on where the potential of zero is defined, the potential difference between any two points in the circuit will be the same regardless of the point selected to be zero potential.
Hope this helps.
A: It's handy conceptually because it allows you to think about the cause of PE separately from the effect (ie the PE itself), which in turn makes it easier to model physical rules in a way that's more generally applicable.
For example, an object gains PE if you raise it above the Earth's surface. If you imagine a cliff 100m high, then ten different objects with ten different masses would gain ten different amounts of PE by moving from the bottom of the cliff to the top. If you ask 'what's the difference in potential energy between the top and the bottom' the answer is that it depends on the mass involved. However, if you work with the concept of potential, you can say that the potential difference between the top and bottom of the cliff is always gh, and you can compare that in a meaningful way with other potential differences. Also, potentials as a function of space or distance crop up in many equations. In quantum mechanics, for example, the Schrödinger equation includes the potential as part of the Hamiltonian.
A: It is similar to a system of coordinates. When we want to know the distance between points in a room it is easy to measure directly. But for the short airline route between two cities we can use the information of latitude and longitude for each, and calculate the distance.
In the case of gravitational potential, knowing the level curves of a map near a river allows to calculate hydroelectric potential for a plant for example. Also points in a electrical circuit with assigned potentials can be used to know available energy for some device.
The advantage of the concept of potentials is to have a bunch of information that can be used to calculate things, that many times were not even in consideration when the mapping was made.
