Is the QED equation of motion for the wavefunction just the Dirac equation? In the QED wiki article, they derive the equation of motion under the Lagrangian, using the classical Euler-Lagrange equation.  This gives
$$(i\gamma^\mu\partial_\mu - m)\psi = e\gamma^\mu A_\mu\psi.$$
Isn't that just the Dirac equation?  Or if not, how is it different?  So then, are the Dirac and QED Lagrangians identical, and the only difference is that QED does the whole path integral instead of just the path of stationary action?
 A: 
Isn't that just the Dirac equation? Or if not, how is it different?

Yes, it is the Dirac equation for a charged fermion interacting with the electromagnetic field.

So then, are the Dirac and QED Lagrangians identical, and the only difference is that QED does the whole path integral instead of just the path of stationary action?

Exactly. The Dirac equation is often mentioned in the context of Relativistic Quantum Mechanics, in which case the wavefunction is a classical field. Meanwhile, QED considers a quantum field, which will lead to more complex corrections. For example, the two theories will yield different values for the electron's anomalous magnetic moment, which was one of the first successes of QED over the simple Dirac theory.

the only difference is that QED does the whole path integral instead of just the path of stationary action

Let me also mention that this is a hell of a difference. It makes the theory incredibly more complicated, since now you are working with quantum fields. It gets much richer from the mathematical point of view ($\psi$ now is not a function of spacetime, but an operator-valued distribution) and QED actually matches experimental data, while the Dirac equation alone doesn't (in today's precision).
