Hi Sumit! Can in your question the electrodynamics be substituted by mechanics:-)?

In classical electrodynamics one solves the classical equations of motion to find the physical states and the density of states is sharply localized on them;  in quantum electrodynamics one takes into account configurations that do not solve the classical equations of motion, in order to obtain the density of states, that isn't sharply localized, but has a non-trivial distribution, determined by the correlation functions. The difference between electrodynamics and mechanics is that the former is a field theory, i.e. a dynamical system of an infinite number of degrees of freedom, whereas the latter describes a finite number of degrees of freedom. 

QED was the earliest relativistic extension of quantum mechanics. Situations in quantum mechanics entail incredibly wild energy oscillations (think of something as simple of electron scattering), and special relativity demands such energy be capable of conversion to matter (consider the textbook examples of Feynman diagrams). QED therefore allows for the "production" of particles/antiparticles, and of course this doesn't even make sense in classical electrodynamics. QED is an extension of quantum mechanics made possible by a mathematical fusion to allow QM's compatibility with special relativity. As both special relativity and QM differ drastically (well, special relativity not so much, but QM does) from classical electrodynamics, I would say the fundamental difference is actually quantum physics. However, if I had to pick an example, it would be that one I gave: the requirement for the production of antiparticles and conversion of mass-energy (and therefore also the meaninglessness of the classical distinction between waves and particles and the change in the law of conservation).

The production of antiparticles is a consequence of special relativity and charge conservation, not, necessarily, a quantum effect-that's a consequence then of the relativistic invariance of the quantum theory. And that it isn't possible to write an interaction term for a finite number of relativistic particles is a theorem in classical mechanics, (though, of course consistent with quantum mechanics), cf. Leutwyler's paper, http://link.springer.com/article/10.1007%2FBF02749856#page-1

Energy-momentum conservation-one of whose ``charges'' is mass, is a consequence  of (global) Lorentz invariance, which is an example of Noether's theorem, that holds for solutions of the classical equations of motion-and is expressed in terms of identities between correlation functions, for field configurations that don't satisfy the classical equations of motion. 

So the difference is taking into account the  contribution of configurations that do not satisfy the classical equations of motion. 

(Indeed the correct way to describe things is that one starts from all possible field configurations, then one finds that  it is possible to  assign to each an appropriate ``weight'', so that one finds that the configurations that are ``typical'' in the ``classical limit'', are those that satisfy the classical equations of motion.)

The fact that antiparticles are a consequence of the mass-energy equivalence of special relativity don't make them any less a part of quantum electrodynamics. The question wasn't "what is the difference between classical electrodynamics and quantum mechanics." Also, QED and QFT more generally grew in particular out of the inability for QM to incorporate special relativity. Classical electrodynamics is not always used consistently but is often used to include classical relativistic electrodynamics (indeed, as Feynman said, "[t]he theory of relativity was developed because it was found experimentally that the phenomena predicted by Maxwell's equations were the same in all inertial systems), and thus special relativity, yet no virtual photons or positrons..

As for reducing electrodynamics to solving "the classical equations of motion", already foreshadowed before Hamilton in Maxwell's A Dynamical Theory we find that "the unifying power of the Lagrangian approach lay in the fact that it ignored the nature of the system and the details of its motion"

Morrison, M. (2000). Unifying Scientific Theories: Physical Concepts and Mathematical Structures. Cambridge University Press.

(see also Siegal's Innovation in Maxwell's Electromagnetic Theory)

Of course, I'm not even sure what "classical equations of motion" you mean (Maxwell's? Hamilton's?), and the conceptual foundations certainly differ from the modern uses of either theory. However, that the difference boils down to this and configurations "that don't solve the classical equations of motion"  is, it seems, to find an important similarity (field configurations) and then to see how these differ in the two theories. Both theories are field theories and thus local theories in the sense Cao refers to in the editor's introductory paper to the proceedings volume Conceptual Foundations of Quantum Field Theory ("It is beyond dispute that the essential spirit of field theories is captured by the concept of locality"). Configuration and equations of motion were important and problematic for both.

Were special relativity trivial to the differences between QED and the classical theory, it wouldn't have been such a problem (involving solutions rejected only to be rediscovered later) in the development of QFT. In QED, electromagnetic forces are ascribed to "virtual photons". Positrons arose mathematically before experimental confirmation due to the requisite mathematical adaptions to QM required to incorporate special relativity. Actually, given the shaky nature of QED, it is hard to even answer this question without some arbitrary divisions between QED and QFT or a selection from various different (in nature and conception) approaches: Dirac's equation, S-matrices, Feynman's rules, etc. It is all rather a jumble, but can be presented much more coherently only thanks to the development of QFT more generally.

Physics is distinct from history of physics-facts themselves matter more than the many equivalent ways of describing the same content. There's no point in repeating the historical difficulties that people had with concepts, but focusing on the understanding that's now available. Of course it's  well known that there is a Hamiltonian and a Lagrangian formulation to classical and quantum physics and that the two formulations are equivalent-this is a mathematical statement, whose proof can be found in any textbook on the subject itself; these are more relevant for technical issues than textbooks on the history of the subject. Which formulation to use is a matter of convenience, not a matter of conceptual significance-it's ``easier'' to do certain calculations in one formulation and/or to recognize certain properties more ``easily''. That's all. The quotation marks, precisely, highlight the fact that these statements express personal taste and nothing more. The technical statement of equivalence means that that it doesn't matter what description one uses. 

For a very nice presentation of how some subtle issues of classical electrodynamics can be resolved, cf. https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf

a paper that's not as well known as it ought to be-though the discussion in, for instance, the first chapter of Itzykson and Zuber's textbook does recall the content. 

Quantum electrodynamics is now understood as part of the Standard Model and the Standard Model is but one quantum field theory and how this can be understood has been for decades the topic of graduate courses. While there are many issues about the Standard Model and quantum field theory more generally,  that are the subject of active research, how to describe interactions and virtual particles isn't one of them-how to do this is well understood, it's background knowledge.  Cf. for instance,https://www.physics.harvard.edu/events/videos/Phys253

and the first chapters of the first volume of S. Weinberg's textbook. 

Well, it may not be a simple answer, but simple terms classical electrodynamics deals with electric and magnetic fields caused by macroscopical distributions of charge and current. As you may know, the framework of this theory is stated by Maxwell Equations. From here, QED took decades of development (involving quantum theory also) and from here cannot be stated a short answer. Paul Dirac extended the background of special relativity in quantum mechanics, and also this was taken as framework for the development of QED for other several scienciests.  QED is the quantized field theory of the electromagnetic force, and applies to all electromagnetic phenomena associated to charged fundamental particles.

Best,

Amenosis

As a subject (such as Physics) evolves with time, theories become more universal in their applicability. For instance now even QED should not be looked at as an isolated theory, it should be viewed on the contrary as a part which is integrated in Glashow-Wienberg-Salam standard model. In classical electrodynamics, as we notice from four Maxwell's equations(1861-62), interactions were thought to be instantaneous. If you have a electric charge distribution, and I have a test charge, then, my test charge should  feel the presence of your charge distribution without any delay in time. Now if you reorder your charge distribution, my test charge should detect the rearrangement process again without any time delay. Such a queer thing happens because of the "action at a distance" principle which is built into the development of classical electrodynamics.

Advent of the relativity theory (Einstein 1905) taught us that a signal cannot travel faster than the speed of light. Therefore one needs a mediator of forces which carries interactions from one space-time point to another at best at the speed of light. This role can be played by the photon (c.f. Photo electric effect,1905). Thus photon could be integrated in a quantum theory of electrodynamics, also valid in the microscopic world,  where it played the role of a quantum field that mediates interaction between quanta of charges. The coupling strength is dictated by the principle of minimal coupling.