In Classical Electrodynamics the physically important quantity is the electromagnetic tensor $F\in\Omega^2(M)$ where $M$ is spacetime.
It turns out that since $dF = 0$ by Poincare's lemma (assuming that $M$ is contractible), there should be $A\in \Omega^1(M)$ such that $F = dA$.
This $A$ is then used to simplify things because it is simpler to compute $A$ and $A$ directly yields $F$. Furthermore, $F$ is more fundamental, since $F$ may exist even if $M$ is not contractible while in this case $A$ is not guaranteed to exist since Poincare's lemma doesn't apply. More than that as I've said, the physical thing is $F$, while $A$ is not. One way to see this is that we can add any $d\phi$ to $A$, because $d(A+d\phi)=dA$ and $F$ is not altered and hence the physics is not changed.
Now let's get to QED. In QED it turns out that it seems (at least by the treatment the books I'm reading use) that the important object is $A$. The field associated to the photon is $A$, the field one quantizes is $A$ and $A$ yields the Feynman rules.
The field $F$ appears in the lagrangian, but it is directly written in terms of $A$ and everything is done with $A$. Hardly ever the field $F$ seems to be used in QED.
Why is that? If even from classical EM, we know that the physical thing is $F$ and $A$ is just something to make life easier in computations that doesn't carry physical meaning itself, why in QED it seems the important object is $A$? In that case, what ends up being the role of $F$ in QED?