If the classical Maxwell theory describes E&M fairly, how well-done is the classical Yang-Mills theory for chromodynamics? If the classical Maxwell theory describes electrodynamics (electromagnetism E&M) fairly well, how suitable would the classical Yang-Mills theory (say SU(3) color) describe the chromodynamics, and how precisely?
I suppose the heavy quarks (c,t,b), being more non-relativistic and we can still heavily use the classical Yang-Mills theory? For light quarks, being relativistic, we may need to consider the classical Yang-Mills + relativistic Dirac equations, but would classical versions of Yang-Mills + relativistic Dirac theories and their Equations of Motions (EOMs) would be suitable to certain levels? How?

p.s. This is a question concerning comparing the
classical v.s. quantum  Maxwell/Yang-Mills theory.
and how applicable are them to describe the real world:
electrodynamics / chromodynamics.

 A: This is nicely answered in Jaffe-Witten's "problem description" of the problem of quantization of Yang-Mills theory:

By the 1950s, when Yang–Mills theory was discovered, it was already
  known that the quantum version of Maxwell theory – known as Quantum
  Electrodynamics or QED – gives an extremely accurate account of
  electromagnetic fields and forces. In fact, QED improved the accuracy
  for certain earlier quantum theory predictions by several orders of
  magnitude, as well as predicting new splittings of energy levels.
So it was natural to inquire whether non-abelian gauge theory
  described other forces in nature, notably the weak force (responsible
  among other things for certain forms of radioactivity) and the strong
  or nuclear force (responsible among other things for the binding of
  protons and neutrons into nuclei). The massless nature of classical
  Yang–Mills waves was a serious obstacle to applying Yang–Mills theory
  to the other forces, for the weak and nuclear forces are short range
  and many of the particles are massive. Hence these phenomena did not
  appear to be associated with long-range fields describing massless
  particles.
In the 1960s and 1970s, physicists overcame these obstacles to the
  physical interpretation of non-abelian gauge theory. In the case of
  the weak force, this was accomplished by the Glashow–Salam–Weinberg
  electroweak theory with gauge group H=SU(2)×U(1). By elaborating the
  theory with an additional “Higgs field”, one avoided the massless
  nature of classical Yang–Mills waves. The Higgs field transforms in a
  two-dimensional representation of HH; its non-zero and approximately
  constant value in the vacuum state reduces the structure group from H
  to a U(1) subgroup (diagonally embedded in SU(2)×U(1). This theory
  describes both the electromagnetic and weak forces, in a more or less
  unified way; because of the reduction of the structure group to U(1),
  the long-range fields are those of electromagnetism only, in accord
  with what we see in nature.
The solution to the problem of massless Yang–Mills fields for the
  strong interactions has a completely different nature. That solution
  did not come from adding fields to Yang–Mills theory, but by
  discovering a remarkable property of the quantum Yang–Mills theory
  itself, that is, of the quantum theory whose classical Lagrangian is
  the Yang-Mills Lagrangian. This property is called “asymptotic
  freedom”. Roughly this means that at short distances the field
  displays quantum behavior very similar to its classical behavior; yet
  at long distances the classical theory is no longer a good guide to
  the quantum behavior of the field.
Asymptotic freedom, together with other experimental and theoretical
  discoveries made in the 1960s and 1970s, made it possible to describe
  the nuclear force by a non-abelian gauge theory in which the gauge
  group is G=SU(3). The additional fields describe, at the classical
  level, “quarks,” which are spin 1/2 objects somewhat analogous to the
  electron, but transforming in the fundamental representation of SU(3).
  The non-abelian gauge theory of the strong force is called Quantum
  Chromodynamics (QCD).
The use of QCD to describe the strong force was motivated by a whole
  series of experimental and theoretical discoveries made in the 1960s
  and 1970s, involving the symmetries and high-energy behavior of the
  strong interactions. But classical non-abelian gauge theory is very
  different from the observed world of strong interactions; for QCD to
  describe the strong force successfully, it must have at the quantum
  level the following three properties, each of which is dramatically
  different from the behavior of the classical theory:
(1) It must have a “mass gap;” namely there must be some constant
  Δ>0  such that every excitation of the vacuum has energy at
  least Δ.
(2) It must have “quark confinement,” that is, even though the theory
  is described in terms of elementary fields, such as the quark fields,
  that transform non-trivially under SU(3), the physical particle states
  – such as the proton, neutron, and pion -- are SU(3)-invariant.
(3) It must have “chiral symmetry breaking,” which means that the
  vacuum is potentially invariant (in the limit, that the quark-bare
  masses vanish) only under a certain subgroup of the full symmetry
  group that acts on the quark fields.
The first point is necessary to explain why the nuclear force is
  strong but short-ranged; the second is needed to explain why we never
  see individual quarks; and the third is needed to account for the
  “current algebra” theory of soft pions that was developed in the
  1960s.
Both experiment – since QCD has numerous successes in confrontation
  with experiment – and computer simulations, carried out since the late
  1970s, have given strong encouragement that QCD does have the
  properties cited above. These properties can be seen, to some extent,
  in theoretical calculations carried out in a variety of highly
  oversimplified models (like strongly coupled lattice gauge theory).
  But they are not fully understood theoretically; there does not exist
  a convincing, whether or not mathematically complete, theoretical
  computation demonstrating any of the three properties in QCD, as
  opposed to a severely simplified truncation of it.

