Mass of the electron In the classical limit, three quarters of the mass-energy of the electron come from the energy of the electromagnetic field of its charge (see Electromagnetic Mass). Intuitively one would expect that the exact formula in the quantum limit may be different, but the general concept would remain, meaning that at least some mass of the electron may be due to its charge.
On the other hand, the main differences between the electron and neutrino are the mass and charge. All other quantum numbers are the same. This fact as well points to an intuitive conclusion that the mass of the electron in part may be due to its charge.
However, from the Standard Model, we learn that the mass of the electron is due to its coupling to the Higgs field. What is the proper way to resolve this apparent contradiction? Does the full mass of the electron come from its interaction with the Higgs field? And if so, what is the connection (intuitive or mathematical) between the classical limit involving the energy of the electromagnetic field and the quantum limit involving the interaction with the Higgs field?
The Wiki link above states:

As to the cause of mass of elementary particles, the Higgs mechanism in the framework of the relativistic Standard Model is currently used. In addition, some problems concerning the electromagnetic mass and self-energy of charged particles are still studied.

Is there any progress or insight into these studies?
A related, but different question: Is the Higgs field needed to explain the mass of the electron?
 A: The electromagnetic interaction only drives the running of the electron mass $m(p^2)$, whereas the starting point of the running $m_0$ is determined by the Higgs mechanism.
Let's first consider the electromagnetic contribution (self energy $\Sigma(\not{p})$) to the electron propagator in the absence of Higgs field,
$$
G = \frac{i}{\not{p}- \Sigma(\not{p})+i\epsilon}.
$$ 
Without renormalization, the quantum loop corrections to the self energy $\Sigma(\not{p})$ would be infinite. However, with proper gauge fixing (Landau gauge) and regularization, it can be shown that the renormalized 
$$
\Sigma(\not{p}=0) = 0
$$
as long as the fine-structure constant satisfies (Maskawa, Toshihide, and Hideo Nakajima, Progress of Theoretical Physics 52.4 (1974): 1326-1354.)
$$
\alpha = \frac{e^2}{\hbar c} < \alpha_{cr},
$$
where non-zero critical coupling is
$$
\alpha_{cr} = \frac{\pi}{3}.
$$
Since we know that electromagnetic interaction is "weak",
$$
\alpha = \frac{1}{137} < \frac{\pi}{3},
$$
it is not strong enough to induce spontaneous axial symmetry breaking (a non-zero $\Sigma(0) \neq 0$ breaks the axial symmetry). Therefore, electron remains massless. Note that the QCD color integration is "strong" enough to induce non-zero masses of quarks (not to be confused with the proton/neutron masses, which are resulted from quark confinement), even though it only contributes a small portion of quark masses compared with those from the Higgs mechanism.
Now enter the Higgs mechanism. The electron propagator in the presence of Higgs field reads
$$
G = \frac{i}{\not{p}-m_0 - \Sigma(\not{p})+i\epsilon},
$$
where electron mass
$$
m_0 = y\upsilon 
$$ 
is dependent on Yukawa coupling $y$ and the non-zero vacuum expectation value $\upsilon = <0|h|0> \neq 0$ (which spontaneously breaks both the electroweak and the axial symmetries) of Higgs filed $h$.
Two comments on the self energy $\Sigma(\not{p})$:


*

*There are quantum corrections to $\Sigma(\not{p})$ stemming from the exchange of Higgs particles in addition to electromagnetic interaction. One can potentially regard the exchange of Higgs particles as the fifth interaction (depending on your definition of fundamental interaction). Nevertheless, this "fifth interaction" is negligible under most circumstances. 

*In the same vein as the Higgsless case, the electromagnetic quantum correction to the renormalized self energy at zero momentum $\not{p} = 0$ is still zero  $\Sigma(\not{p} = 0) = 0$ (I am not aware of any prove, anyone could point to any references?), given the "weakness" of the electromagnetic interaction. In other words, the mass $m_0 = y\upsilon $ is solely coming from the Higgs vacuum expectation value $\upsilon = <0|h|0>$. That being said, the physical mass $m_p$ (define by the pole of the electron propagator at mass shell) will receive a finite quantum correction arising from $\Sigma(\not{p}) - \Sigma(0) \neq 0$. This quantum contribution might be the "roughly 20% of the mass of the electron" Polchinski is referring to (mentioned in Mitchell Porter's answer as the renormalization group running of the mass). The electromagnetic interaction only drives the running of the electron mass $m(p^2)$, whereas the starting point of the running $m_0$ is determined by the Higgs mechanism. 

A: The renormalization group running of the mass in quantum field theory, is the quantum version of the classical self-energy contribution. According to Polchinski's String Theory, section 16.2, "the self-energy... is roughly 20% of the mass of the electron". So even in the standard model, the yukawa coupling to the Higgs field is the dominant contribution to the electron mass, but it isn't everything. 
This also suggests that if the mechanism of neutrino mass were exactly the same as this, but with no contribution from electromagnetic self-energy, it would still be of the same order of magnitude as the electron mass. And indeed, the typical beyond-standard-model theory explains the tiny neutrino mass quite differently, as due to a "seesaw mechanism" in which a superheavy Majorana mass and an electroweak-scale Dirac mass combine to produce a superlight observed mass. 
Theories which would explain the neutrino mass without the seesaw, like the "neutrino minimal standard model", have to suppose a neutrino yukawa coupling that just happens to be many orders of magnitude smaller than any other yukawa couplings. I am not aware of any model in which the difference in masses can be attributed to the difference in charge and corresponding difference in self-energy.  
