What's the difference between an electron and a quark? Could you explain how QFT defines the peculiarity and differences, as to charge, between an electron and a quark?

*

*A quark has a fractional charge.  Is the Coulomb force proportional to these charges?  What is the (value of the) repulsive force between two $u$ quarks?

*What is the (value of the) attractive force between a $u$-quark and a $d$-quark? In this case does the gluon act with repulsive force?


Note: I have focused only on the aspect of charge and luckily the question has been reopened, please complete your previous answers, correct eventual mistakes or misconceptions, what escapes me in particular is the emission of an electron in beta decay: if the charge changes from udd to udu and is balanced, how is the formation of a new charge -3/3 explained? where does it come for?
 A: For a better understanding you also need to add the neutrino ($\nu$) to the picture.
Thus I will compare the leptons ($\nu$ and $e$) on one hand
with the quarks ($u$ and $d$) on the other hand.
Leptons:
$$\begin{array}{c|c|c|c}
 \text{} & \text{spin} & \text{charge} & \text{weak isospin} \\ \hline
 \nu & \frac 12 & 0  & +\frac 12 \\ \hline
 e   & \frac 12 & -1 & -\frac 12  
\end{array}$$
Quarks:
$$\begin{array}{c|c|c|c}
 \text{} & \text{spin} & \text{charge} & \text{weak isospin} & \text{color} \\ \hline
 u & \frac 12 & +\frac 23 & +\frac 12 & r,g\text{ or }b \\ \hline
 d & \frac 12 & -\frac 13 & -\frac 12 & r,g\text{ or }b
\end{array}$$
Let us look at the differences.

*

*The charges of leptons and quarks differ "slightly" (by a $\frac 23$ difference).


*Regarding the weak interaction (see the "weak isospin" column
in the tables aboves) there is no essential difference
between leptons ($\nu$ and $e$) and quarks ($u$ and $d$).


*The main difference between leptons and quarks is that quarks have
an additional property named "color".
Every quark comes in three different colors (red, green or blue).
So instead of just $u$ quarks there are actually three different kinds
of them ($u_r$, $u_g$, $u_b$). And similarly for $d$ quarks.
Much like charged particles interact via the electromagnetic interaction,
colored particles interact via the strong interaction.
That is why several quarks combine to form composite particles
(like protons, neutrons, pions, ...).
Leptons don't do this because they are colorless.
A: The “Standard Model” of particle physics recognizes four fundamental interactions, and describes how different fields participate in them.
The theory of fields is called “quantum” because fields can only exchange energy, momentum, or angular momentum with each other in lumps.  In particular, angular momentum can only be exchanged in lumps with size $\hbar$.
It is sometimes convenient to refer to such a lump as a “particle.”
Fields whose associated angular momentum is an integer multiple of $\hbar$ can be thought of as “force carriers”. Fields whose associated angular momentum is $\hbar/2$ can be thought of as “matter particles.” Each field also has an associated electric charge.
The phrase “intrinsic angular momentum” or “associated angular momentum” has too many syllables for a normal person to say, so we call it “spin,” even though that conjures up an unphysical picture of a spinning ball.
The interactions and their participants are:

*

*Gravity, the weakest, felt by all forms of mass-energy.  Our successful theory of gravity is a curved space-time background against which the other fields develop.  If there is a particle associated with gravity, the symmetries involve suggest such a “graviton” would have spin $2\hbar$.


*The “weak nuclear interaction,” which has three associated charge states.  The “charged current” is mediated by the $W^\pm$, and connects particles in the six “flavor doublets” of the Standard Model, such as $u\to Wd$ or $e\to W\nu$.
The $W$ has unit spin and unit charge, so the flavor doublets must have spin $\hbar/2$ and a charge difference of one unit.
All of the twelve known matter particles also interact with the “neutral current,” mediated by the $Z$.  Search my posting history for “weak charge” for comments on the neutral current.


*The “electromagnetic interaction,” mediated by the photon.
The electric charge is said to have $U(1)$ symmetry, which vaguely means it can be positive, negative, or zero.
Neutrinos do not participate in electromagnetism, because they have zero electric charge.


*The “strong nuclear interaction,” mediated by gluons.  The associated “color charge” is said to have $SU(3)$ symmetry, which means it can be positive, negative, or zero in three different ”directions,” commonly labeled “red-antired” and two other color-anticolors.  The charged leptons, including the electron, do not participate in the strong interaction, because they have zero color charge.
(The Higgs field mediates a sort of self-interaction, so I leave it out of this list.)
Because the color interaction is strong, quarks are always found as “color singlets.” Such an agglomeration can consist of a quark-antiquark pair, or of three matter quarks whose same-sign colors combine to zero.  A quark-antiquark pair is a “meson,” and a three-quark system is a “baryon.”
So in broad strokes, the difference between an electron and a quark is that electrons are “color singlets” all by themselves, and ignore an interaction in which the quarks participate.
A: 
Could you explain how QFT defines the differences between an electron and a quark?

Quantum Field Theories are mathematical tools used to model data and observations in various disciplines of physics, from particle physics nuclear physics, etc

A quark has just a fractional multiple charge but apparently behaves in different ways,

Quarks and electrons are elementary particles in the standard model of particle physics

can you list and justify them?

The list  is in the table in detail, .

and it cannot be theoretically justified because it is axiomatic for the standard model field theory. It is the result of a laborious process of  studying the symmetries in particle scatterings and production over almost a century,  that led to the quark model.
The standard model accepts  that there are three forces with  which elementary particles interact, the strong which is the realm of quarks, the weak which is the realm of quarks and leptons, and the electromagnetic which rules where charges exist. To understand it one has to study the particular field theory for elementary particles that the SM uses.
In comments:

Am I right to say: suppose we can't say that a quark is just en electron with a slightly smaller charge,? –

The different interactions of the different particles are what defines their difference, as the bold in the last sentence points out. A quark interacts with the strong , weak and electromagnetic, the electron only with electromagnetic and weak.
A: The Quark model developed by George Zweig and by Murray Gell-Mann is just a classification scheme for hadrons, it is not a field theory. According to this model, all hadrons are "made" of smaller elements called quarks (see famous article "Structure of the proton" by Feynman in Science, 1974, for more details). Quarks in this model have angular momentums (spin) $\frac{1}{2}$, just like electrons.
The most advanced field theory describing quarks is called quantum chromodynamics (QCD). Quarks in QCD are described by the wave functions (first rank spinors) similar to those used for electrons in quantum electrodynamics (QED). "Similar" means that wave functions transform in the same way under Lorentz transformations and have the same number of components. Wave functions of this sort are usually used to describe particles with spin $\frac{1}{2}$.
The difference between electrons in QED and quarks in QCD is that electron fields are coupled to electromagnetic field only, while the quark fields are also coupled to some other, more complicated, fields. Hence, of course, quarks and electrons behave in different ways in QFT.
