What is problematic about the Moving magnet and conductor problem? The problem is posed as follows:
There is a conductor and a magnet in relative motion. This motion induces emf in the conductor. The value of the induced emf is independent of whether it's the conductor or the magnet that is moving. 
This phenomenon has two different explanations depending on the frame of reference one is in. If one is in a frame at rest with respect to the conductor and the magnet is moving, the induced emf results from the variation in the magnetic flux which produces electric filed as dictated by Faraday's law.
On the other hand, if one is in a frame that is at rest with respect to the magnet and the conductor is moving, then the induced emf is a result of the lorentz force acting on the charge carries moving in a magnetic field.
This problem was one of the motivations that led Einstein to develop SR:

It is known that Maxwell's electrodynamics – as usually understood at
  the present time – when applied to moving bodies, leads to asymmetries
  which do not appear to be inherent in the phenomena. Take, for
  example, the reciprocal electrodynamic action of a magnet and a
  conductor. The observable phenomenon here depends only on the relative
  motion of the conductor and the magnet, whereas the customary view
  draws a sharp distinction between the two cases in which either the
  one or the other of these bodies is in motion. For if the magnet is in
  motion and the conductor at rest, there arises in the neighborhood of
  the magnet an electric field with a certain definite energy, producing
  a current at the places where parts of the conductor are situated. But
  if the magnet is stationary and the conductor in motion, no electric
  field arises in the neighborhood of the magnet. In the conductor,
  however, we find an electromotive force, to which in itself there is
  no corresponding energy, but which gives rise – assuming equality of
  relative motion in the two cases discussed – to electric currents of
  the same path and intensity as those produced by the electric forces
  in the former case.

So yeah, We have physical situation that depends only on the relative motion, But the laws of physics draw a distinction between what is moving and what is at rest. 
But I still can not understand what is problematic about this situation? 
What is wrong about using two different laws of physics(Lorentz force and Faraday's) depending on one's frame of reference to describe the same phenomenon?
 A: Nothing is problematic with it.
As FraSchelle says above—and is also true in the development of many other physics theories, in that they are, over time, purified of the scaffolding that helped construct them*—the original motivation doesn't affect the content of the developed theory.
*cf. the top of p. 90 (PDF p. 91) of Stefano Bordoni's When Historiography met Epistemology: Pierre Duhem's early philosophy of science in context for some historical examples
Einstein, when discussing the "asymmetries which do not appear to be inherent in the phenomena," was referring to a particular interpretation of Lorentz's electrodynamics that he learned, during 1896-1900, from an 1894 book by Föppl.


*

*See §15.1.6, "Origin of the Asymmetry Pointed Out by Einstein," pp.
264-265 (PDF pp. 284-285) of André K. T. Assis's free
Relational Mechanics and Implementation of Mach's Principle with
Weber's Gravitational Force.


*

*See Faraday's, Maxwell's, and Weber's explanation of this phenomenen in the passages quoted on ibid. pp. 258-264 (PDF pp. 278-284).


A: As I understand the paragraph, Einstein is looking for missing energy.  In the case of a moving magnet, there is an induced electric field having energy.  See, for example The Feynman Lectures Vol. II Chapter 8, Section 5 Energy in the electrostatic field.  But when the wire moves toward the magnet, there is an electromotive force which is not due to an electric field, and therefore no electric field energy.
Thus in the first scenario, Maxwell's theory states that there is more energy in the system than there is in the second scenario.  But they should both be describing the same system from different perspectives, and should therefore give the same energy value.
It seems worth noting that there is an energy attributable to the (static) magnetic field, but that is present in both scenarios.
