Why do we talk more about electric potential in electrostatics and not electric potential energy? There is a concept in electrostatics called electric potential, which is defined as the amount of electric potential energy (which is a clear translation of gravitational potential energy except for minor differences like charges - but two opposing charges basically act just like gravity) per coulomb. 
Electric potential seems to be more used to describe charges in electric fields in electrostatics more than electric potential energy, but this doesn't seem to be the case for gravitation, there's just gravitational potential energy, not a gravitational potential. Why is that? 
Why not just talk about electric potential energy in electrostatics? Why divide by charge?
And on another note, what I want to get at most is: how is electric potential and electric potential energy fundamentally different (neglecting formulae)?
 A: The concepts of potential and potential energy are both used for both electric fields and gravitational fields.  In light of some of the existing answers to your question, I think we all now understand that.  The remaining question is why we talk about the potential more often when discussing electricity and the potential energy more often when discussing gravity.
In my opinion this is a habit formed by many of us because of the nature of the basic problems and early experiments in the respective areas.
The "discovery" of gravity was in the context of falling objects and orbiting planets, which is also the context of most introductory problems involving gravity.  In those contexts, one is discussing a specific, identifiable object within the field, so it is natural to consider the potential energy relevant to that object.
For electricity, the topics early in its study are about currents and batteries where one is discussing a flow of many charged particles, no one of which is particularly important (or visible).  Even when studying static electricity with charged metal objects, the charge often cannot be treated as a single entity, since it moves about within the object to equalize the internal potential.  So we mostly talk about potential because it is the more convenient concept.
A: I cannot answer why it is more common to talk about electric potential than electric potential energy; I think it is a matter of convenience since the electric field is negative the gradient of the electric potential, not the potential energy, and it seems to me that in electrostatics the electric field is more sought after than the electric force (which would be negative the gradient of the electric potential energy).
I do want to emphasize, though, that there is a gravitational analogue to electric potential. You can think of mass as gravitational "charge". Then, the gravitational potential is just gravitational potential energy divided by mass. 
A: We always talk about electric potential (voltage) because it is easily
measured without disturbing the system.    Electric potential energy,
on the other hand, requires global knowledge (not just two probes
with a one-second measurement), because that energy requires knowledge of all the charges present
in the field.    Those charges could be hard to hunt down
(and measuring the charge in, for instance, a capacitor will
require fully charging and discharging the capacitor to calibrate
it).   
To measure the voltage in a Leyden jar, it suffices to watch an
electroscope foil take on a slant.   To measure the energy, 
is traditionally done by discharging through multiple persons
in series, and seeing how high they jump.
Voltage is an intensive property, it is measurable with a small
local probe.   Potential energy is an extensive property, it requires 
an assessment of many disparate parts.
A: Nice question.
First of all, the IS a gravitational potential. I use to call it $V_g$ or $V_{\vec{g}}$, while the electric potential is $V_E$. "Potential" itself is as used as "potential energy", at least in my surrounds.
Secondly, there is a high issue with this notation. Many books decide to write $\phi$ for the potential (Greek letters for scalars, but that's absurd because $W$...$U$... are also scalars). Then, many books call kinetic energy $T$, and potential energy $V$. This is very upsetting to me, because it has just conquered everything, and the logical notations $E_k$ and $E_p$ have been just deposed. For me $T$ will always be period, and $V$ always volume (but I've got used to the other one, I can work with it without problems).
What I'm trying to say in this apparently off-topic paragraph is that all this has lead many people to say $H=T+V$, and at the same time"The particle is under a potential $V$". In other words, many people say potential when they mean "potential energy", for lazyness and also because of this notation. They know what they are doing, but they say it wrong. They're sometimes mixing concepts in speech (altough they understand it correctly). Students use to suffer this so much.
And finally, the potential itself is more used because it doesn't directly depend on your system. I mean, you can have an empty space with a certain potential. When you place the charge, the charge will get the corresponding potential energy. Now, there are so many systems in which you can somehow keep the potential constant. 
For example, a battery supplies a constant potential to a conductor. In fact, in lab you do not have any source of potential energy, but a voltage source. If the system is such that charges do not affect the source of potential (or not significantly), then you have the same potential, but you can add many charges, the total potential energy is different and the potential can be constant.
A: This question is basically boiling down to about the difference of interpretations between electric field $E=kq_k/{r_k}^2$ and electric force $F=kq_iq_j/r_{ij}^2$
Basically, the electric field $$\vec E=\sum_l k\frac{q_l\hat r_l}{r_l^2}$$ tells us how a test charge (with charge +1) would have behaved under such a field, without disturbing the field (in other words, we do not put the test charge into the field). It is something able to tell us local properties in space. Meanwhile electric force can't (in principle), as it is not a field - such forces only tell us about how two objects interact at some specific locations, and they will always disturb each other. 
The significant difference between force and field raises when the field source (e.g. a point charge) oscillates really fast - with the idea "field", you will see waves, as well as the energy goes into space. On the other hand, you will have a hard time seeing such things with only the idea "force", as any object being acted won't react instantly.
Therefore, it is perfectly reasonable to make up something similar in gravitation too (why not, after all?). In fact, our little friend $gh\approx GM/R_{earth}$ from high school physics  is a gravitational potential. ($h$ is the height from surface of earth plus the radius of earth)  
