It is a well-known fact that Klein-Gordon scalar $\Psi(x)$,
$$
(\partial^{2} + m^2) \Psi (x) = 0
$$
as well as 4-vector $A_{\mu}(x)$,
$$
(\partial^{2} + m^{2})A_{\mu} = 0,\quad \partial_{\mu}A^{\mu} = 0,
$$
(and even function of an arbitrary integer spin) describe the field: first, there aren't positive definite norm (with Lorentz invariant fullspace integral) for this functions, and the second, the free solutions are represented in a form of independent harmonic oscillators, like for case of classical electromagnetic field. So we naturally assume commutation relations for amplitude operators of these fields.
Then let's have the Dirac equation and corresponding function (in general – let's see the function of arbitrary half-integer spin). Let's also assume, that we don't know that it describes some particle. We can build positive definite norm (with Lorentz invariant fullspace integral), and the solution for field also looks like harmonic oscillator. But for positive definite of energy we must assume anticommutation relations.
So, the question: why do we assume that Dirac spinor $\Psi$ (or, in general, tensors of an arbitrary spin) describes only the particle, not the field? In my opinion, the fact about positive definite norm leaves the possibility for the description of the field by this spinor (not the particle).
My question is not about formal definition of these functions. Of course, all of them are relativistic fields. But they describe different physical objects in classical limit – fields and particles correspondingly. Maxwell function $A_{\mu}$ describes the EM field even in classical limit, but the Dirac spinor $\Psi$ describes the electron only in the quantum case (when QM postulates work).
Best Answer
In QFT, the Dirac spinor will also be promoted to a field, whose oscillation mode coefficients are creation and annihilation operators.
BUT: For the Dirac spinor it is possible to well-define a probablility density and current:
$$\rho^\mu \propto \bar\psi \gamma^\mu \psi$$
This current's zero component is positive definite and using the Dirac equation one can show that it is conserved, i.e. $\partial_\mu \rho^\mu \equiv 0$.
Therefore, besides being interpreted as a quantum field, the Dirac spinor can be interpreted as a particle wavefunction in regular QM.
Let me remind you, though, that the energy eigenvalues of the Dirac operator are not bounded from below. This is not as problematic, if one agrees to the concept of the Dirac sea of electrons already occupying all negative energy states. While the construction of the Dirac sea is very hand-waving, it provides a key prediction: particle-antiparticle pair creation from 'pure energy' (i.e. a photon).