Aharonov-Bohm Effect and Flux Quantization in superconductors
According to Wigner, the wave function of a quantum particle can be multivalued, i.e., can acquire a nontrivial phase around a closed loop. A phase is nontrivial when it cannot be removed using a gauge transformation by $e^{i \alpha(\theta)}$, with a true function $\alpha$, i.e., $\alpha(2\pi) = \alpha(0)$. The wave functions having this property are sections of nontrivial line bundles over the configuration manifold.
The reason that a wave function is not required to be a true function is because its overall phase and magnitude are nonphysical, if one defines quantum expectations as:$$ \left< X \right>~~=~~ \frac{\int \Psi \hat{X} \Psi}{\int \Psi \Psi} \,.$$
Such wave functions arise when the configuration manifold is not simply connected with a nontrivial cohomology group $\mathcal{H}^{1}(M,\mathbb{R})$ (This is the case of the circle). In this case, there will exist vector potentials on the manifold which are not the gradients of a true function on the manifold. $A \ne d\alpha(\theta)$. with, $\alpha(2\pi) =\alpha(0)$. However, there is no need for the flux to be quantized as the wave function needs not be a true function on the configuration manifold. On the contrary, if the flux had been quantized, then no Aharonov-Bohm effect would not have observed. A quantization condition occurs when $\mathcal{H}^{2}(M,\mathbb{R})$ (The Dirac quantization condition), but this is the case of a particle moving on a sphere rather than on a circle.
However, this is not the case in superconductivity. The difference between the two situations lies in the fact that the "macroscopic wave function" of a superconductor is not a "wave function". i.e., it is not the coordinate representation of a state vector in a Hilbert space. It is a quantum field describing Goldstone bosons (Cooper pair) of the superconducting phase (usually called an order parameter). The modulus of the macroscopic wave function $\left|\Psi \left( \theta \right) \right|^2$ describes the number density operator of the Goldstone bosons. Its two point functions describe the (long range) correlations. This quantum field couples minimally to electromagnetism, and this is the reason why its equation of motion is similar to the Schrodinger equation of a particle coupled to electromagnetism. But the main difference this field is a true scalar field and not a section of a line bundle. This gives us the reason why the phase it acquires in a full loop should vanish because otherwise for example, its correlation functions would depend on how many times the circle was wrapped.
While both the answers given in some sense are correct, the true reason has to do with energetic considerations. It is a matter of what is stronger and can be phrased as the following question: Will the wavefunction alter itself to accommodate the flux, or will the flux quantize itself because the wavefunction is trying to remain single valued?
As an example of what I mean: flux is quantized in the superconducting case up to a point. One can ramp up the flux inside the superconducting ring until superconductivity is destroyed (even though the magnetic field itself is not in contact with the superconductor). This will occur purely because the superconducting condensate will not have enough energy to keep the flux quantized at large values of the flux.
It is because the superconductor in some sense can be thought of as a macroscopic wavefunction with a lot of energy that the flux is quantized in the superconducting case. In the Aharonov-Bohm case, we have a single electron (or a beam of incoherent electrons), which does not have enough energy to alter the flux.
Just adding to @Xcheckr's answer, which I think is the most correct: quantum fields are always single-valued. In a superconductor, it is energetically favorable to minimize the kinetic term $|D_A\psi|^2$, where $\psi$ is the superconducting order parameter. $D_A\psi=0$ implies that the phase of $\psi$ is determined through parallel transport by exponentiating $iq\int A$, and this together with the single-valuedness of $\psi$ enforces flux quantization.
In an AB effect setup by contrast, there is no energetic reason to set $D_A\psi=0$, and so the phase of $\psi(x)$ will not be determined by $\exp(iq\int A)$. This means that for some generic value of the flux, the magnitude $|\psi|$ will not be constant (e.g. it will pass through zero at some point), which is where the interference in the AB effect comes from. In a superconductor $|\psi|$ must be constant for energetic reasons, and this is why the flux is quantized in a SC.