A somewhat formal answer might run as follows: how did we get QED in the first place? We took coordinates for which the EM field behaved as a set of uncoupled oscillators (modes) and then quantized these oscillators. The fact that we do not, usually, look at the wave function of these infinitely many oscillators, is probably more of a mathematical issue, or maybe even a historical accident. There is no intrinsic difficulty in doing so, at least if you do not give undue weight to the unfortunate fact that there are infinitely many oscillators involved.

Look at it as we may, however, we are essentially dealing with a large number of oscillators. If we then look at appropriate Gaussian packets (coherent states), we know that they follow dynamics determined exactly and uniquely by the corresponding classical dynamics. But the EM oscillators were so constructed, that their classical dynamics corresponds exactly to classical Electromagnetism. In this way, it is clear how, at least, certain states in QED will have a dynamics determined by---and in the limit of large amplitudes, close to---the behaviour determined by classical Electrodynamics.

Can I make a similar statement for all states? No: ever since Schroedinger gave us his ``cat'', we have known that quantum mechanics contains states which are delocalized in terms of the classical variables, and which thus do not tend to a well defined limit in the semiclassical limit. Decoherence will presumably get rid of this effect when the state comes in contact with an environment, but this is clearly another story.

You are right.  The EM field can be viewed as a photon wavefunction.

For a photon there can only be a quantum field theory of electromagnetic radiation, there is no quantum mechanics of a photon. The reason being is that photons can never be non-relativistic and can be freely emitted and absorbed hence no photon number conservation.

Yasser, here is a view from a bit different angle when we talk about connections between QM and EM-theory.  A Schrodinger eqn for a wave function of a single particle is congruent with a so called paraxial wave eqn (an appropriate approximation of the Maxwell eqns) in optics/electrodynamics for slowly diffracting beams of light (i. e. when diffraction angle is much smaller than pi/2) -- if one replaces time "t" in Schr. eqn by the spacial axis "z" along which a diffracting beam propagates. They have exactly the same eigen-solutions ,  and the Gaussian solution for a diffracting beam as it propagates in "z"  is the same as a dispersion of a wave function from its initial Gaussian spacial profile with time "t".  A physical entity in EM-wave equation that corresponds to a potential in QM is a space-dependent, slowly varying dielectric constant. (This analogy is well known; it can be found in many textbooks on optics and quantum electronics, e. g. Yariv's books.)

A bit less known fact is that this analogy/congruence breaks down when we go into more general case -- to regular Maxwell equations for arbitrary field geometry in EM-theory vs relativistic equation for wave function in QM. The Maxwell equations produce a "regular" wave equation (laplacian of field - second derivative over time = 0), whereas a relativistic QM equation has a non-zero right hand part -- the so called Klein-Gordon equation. An interesting thing is that that one still has its counterpart  in EM-theory.  In term of electrodynamics it corresponds to a full-blown (no frequency-dependence) wave equation for plasma, whereby a "plasma wavelength" would be now the Compton wavelength, lambda_C=h/(m_0 c)

  First i suggest in order to compare classical and quantum electrodynamic ( or electromagnetism) it is better to study the fields of 4-vector potential A instead of E and B (i.e. electric and magnetic vector fields) because for instance in the case of Aharanov-Bohm experiment the role of vector potential explains the related paradox, where E and B vanishes while vector potential remains non-zero.Description of classic electrodynamics through QED,as mentioned in your question, is simpler than the converse approach. I mean the transition from classic to quantum electrodynamics requires to change representation from space coordinate x to momentum representation,and then by imposing the periodicity conditions of ( wave) we arrive the momentum components (i.e. wave vector k) with discrete values like : kx =2πnx /L etc. here L is the length of periodicity and n in an integer, therefor we attributed  every wave vector to allowed discrete values ( i.e. quantized momentum). in this process we quantized the waves,and the vector potential can be calculated by a sum over all discrete values of  wave vector k. Nevertheless for transition from quantum expression of A , simply we should substitute the summation with an integral calculated over all possible continuous wave vector. In the QED approach finally we arrive to Bose-Einstein statistics regarding photons fields (i.e. vector potential A as photon fields) with definite polarization, so an electromagnetic field representation of photons could be fulfilled by a state of |Nkλ> with λ as polarization and k as wave vector, this may be accomplished by consecutive products of creation operators in second quantization. So the key difference will appear in discrete values of momentum (wave vector) as well as the vector potential representation by a summation over these discrete values.

A somewhat formal answer might run as follows: how did we get QED in the first place? We took coordinates for which the EM field behaved as a set of uncoupled oscillators (modes) and then quantized these oscillators. The fact that we do not, usually, look at the wave function of these infinitely many oscillators, is probably more of a mathematical issue, or maybe even a historical accident. There is no intrinsic difficulty in doing so, at least if you do not give undue weight to the unfortunate fact that there are infinitely many oscillators involved.

Look at it as we may, however, we are essentially dealing with a large number of oscillators. If we then look at appropriate Gaussian packets (coherent states), we know that they follow dynamics determined exactly and uniquely by the corresponding classical dynamics. But the EM oscillators were so constructed, that their classical dynamics corresponds exactly to classical Electromagnetism. In this way, it is clear how, at least, certain states in QED will have a dynamics determined by---and in the limit of large amplitudes, close to---the behaviour determined by classical Electrodynamics.

Can I make a similar statement for all states? No: ever since Schroedinger gave us his ``cat'', we have known that quantum mechanics contains states which are delocalized in terms of the classical variables, and which thus do not tend to a well defined limit in the semiclassical limit. Decoherence will presumably get rid of this effect when the state comes in contact with an environment, but this is clearly another story.

As far as I know, the term "particle" in the picture of quantized fields refers merely to the fact which energy energy eigenstate the field is in at the moment.  The statement "there are n photons with the frequency omega" means that the oscillation mode omega of the field is in the energy eigenstate n .  Thus in my opinion, strictly spoken there are no "particles" at all and no wave/particle duality in field theory. Field is fundamental, particles are the result of their quantization.

Years ago I was wondering, how it is possible that coulomb interactio can be interpreted as virtuel photon exchange. I thought "the electric field is perpendicular to direction of propagation, how can large number of photons exhanged between the charges can produce an effective macroscopic electric field directed from one charge to the other. 

Later, I read a book where I discovered that unlike the Maxwell equations, the covariant field equations of electromagnetic field have solutions with longitidunal and timelike polarizations. It is the integration over these modes that gives the coulomb field. Thus field is the fundamental entity in this picture.