What is the difference between the potential difference and potential energy of an electron? What is the difference between the potential difference and potential energy of an electron?
Let's take an example the potential difference (PD) across a resistor. if there's a current flowing, the power lost is equal to the current x PD. But in reality, the electrons inside the resistor are losing this potential energy right? If so, what is the relationship between the energy level (or the quantised energy) of an electron inside the resistor and voltage drop?
 A: 
What is the difference between the potential difference and potential energy of an electron?

If I understand your question right, these terms are describing the same thing - one is just in a "per charge" version.


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*Electric potential energy $U_e$ is the potential energy associated with one spot in the circuit.

*Electric potential or just potential $V$ is this energy per charge, $V=U/q$.

*Potential difference $\Delta V$ is then just the difference in potential between two points, $\Delta V=V_{b}-V_{a}$.
Potential energy is the "repulsion" (like pressure in a water pipe) that pushes electrons through the resistor.
A: Potential difference in a circuit is a collective effect of a large number (order of 10^23) electrons moving in conductors and resistors . It is a term within classical electrodynamics, and the value emerges from what happens at the level of particles and atoms, the quantum mechanical level.
This link shows what is going on at the microscopic level:

In a conductor the electrons exist in practically continuous energy levels that make them act as free, they have a drift velocity, much smaller than the velocity of light which is the velocity of the collective effect,

this shows more detail of how resistance is seen by individual electrons.
The individual electrons "see" the  potential difference, i.e. the imposed electric field at the classical level and acquire their drift velocity, added to the statistical velocity of the fermi level. 
Reading up on the band theory of solids will help to clarify this microscopic picture, as electrons in circuits are not tied to atoms but to the lattice as a whole.
