what potential difference actually do indicate how much energy each coulomb have, or make current flow, or do both?
The answer is conceptually both.

The key is in understanding that there are three interchangeable ways to describe the force a test charge will feel in a system of other charges:


Distribution of charges.
Electric field.
Electric potential.


These are three different mathematical approaches to describing the same situation, which I've elaborated on extensively in the Voltage and Current section of Ultimate Electronics.

The third of these, electric potential, describes potential energy (per Coulomb). The second of these, electric field, describes the force that will cause current to flow, reaching a force balance described macroscopically as Ohm's Law.

The potential is the integral of the field. The field is the gradient of the potential. They are two descriptions of the same thing, so "both" is the answer that makes the most sense.

An analogy to gravity makes sense too. We could describe a gravitational potential that a mass moves through, proportional to height. Alternatively, we could say that the difference in gravitational potential between height A and height B causes a force, which if unopposed, will cause the mass to move. The missing piece in the gravitational case is something like a linear Ohm's Law resistance which relates force and velocity.

(Note: for simplicity, this entire answer assumes no time-varying magnetic fields.)