The way you can, and must, distinguish between a superposition and a classical probability distribution is that you must do something that causes quantum interference between its terms or, to say another way, you must ultimately do a measurement in a vector basis that is incompatible (the operator it allies with does not commute with) to the one you are interested in.
The mathematical formalism of quantum mechanics can be considered in many ways as a generalization of probability theory: such a concept is called in foundational papers a generalized probability theory or GPT, and we can thus make a point-by-point comparison with classical probability theory. In particular, the analogy of an ordinary probability density function is the quantum state, and the analogy of the Bayesian probability update is "measurement". Note that if you take the GPT view, this pretty much fixes the interpretational issues: states are indeed knowledge/information states and "wave function collapse" is indeed proximately nothing more than a subjective information acquisition. The two terms occupy mathematically exactly the same points in their respective theories.
What makes things "interesting" is their behavior, because after all, it is a generalized probability theory. You see, when you have a classical probability density function or pdf, like say the usual "bell curve"
$$f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$
it is implicitly understood that it only provides information, or models an agent's held information rather, regarding the variable $X$, and any functions of $X$ alone. On the other hand, a wave function, which by the way is only the extreme case of quantum states (and in non-Hilbert formalisms, the distinction doesn't exist explicitly, but it's still useful for analysis), does not just carry information about a single variable $X$, but also carries information about at least one more variable, e.g. in the case of a moving free particle, we also have its momentum, $\mathbf{p}$, determined by the same wave function but in a different way than, say, a mere function of the classical $X$ above would, such that the wave functions that localize $X$ to a point are mutually exclusive with those that localize $\mathbf{p}$ to a point, so no information gain can ever reduce your wave function state to one where both are localized on a point. In a sense, squeezing information into one squeezes it out of the other, and conversely. In particular, if we instead have
$$\psi_x(x) = \frac{1}{(2\pi)^{1/4}} e^{-x^2/4}$$
and we receive information that $x = x_0$, we must change to
$$\psi'_x(x) = \delta^3(x - x_0)$$
but if we receive that $p = p_0$, we must change to
$$\psi'_x(x) = e^{ip_0 x/\hbar}.$$
But note! If we were not allowed to measure $p$, or better, $p$ "didn't exist", and our scenario only included $x$, then in fact the behavior of $\psi_x$ is identical to classical probability theory, and since the proper "probability" is it norm-squared, we can apply this theory to every single instance we can apply classical probability to without any issue at all! It just becomes oddly redundant/complicated, with the additional complex phase term (not invoked in the above).
Thus we can return to your question. For a classical Brownian motion system of two wells, you are right, we can set up a suitable setup in which an agent using classical probability theory is justified in describing its knowledge via
$$f_x(x) = \frac{\mathrm{rect}\left(x - 2\right) + \mathrm{rect}\left(x + 2\right)}{2}$$
where $\mathrm{rect}$ is the "rectangle" or "unit pulse" function, a uniform discontinuous step one unit wide and high, zero everywhere else, and we imagine it as lying in a pair of potential wells of suitable width sunk around $x$-coordinates $\pm 2$. With it energetic enough, it flits over the barrier and produces probability in both. There is also mathematically NO difference between this and the wave function
$$\psi_x(x) = \frac{\mathrm{rect}\left(x - 2\right) + \mathrm{rect}\left(x + 2\right)}{\sqrt{2}}$$
so long as we are only measuring $x$ and functions of $x$ alone!
The problem is, in a real quantum setup, we can also measure $p$.
And then, there's a difference. In fact, it gets even weirder - we can suitably construct a hybrid observable, $q$, that is a function of both $x$ and $p$ such that if $q$ takes a certain value, the "update rule" must set your $\psi_x$ to the above function. In particular, when it already is the above function, that observable must yield that answer with probability 100%!
Now think about that in light of the classical Brownian system. What that is saying is equivalent to saying that although the Brownian particle is only in one well or another at any given time, we had a measurement we could run that would be equivalent to affirming that its probability distribution for all time, given only its state at that single moment in time, 100% for sure, is some certain way! How would you do that? If you just know right now it is dead center of the left well, say, how can you measure just from that information alone that its probability density is the above shape and not some other shape? Remember that probability density only appears in aggregate repetition, not just one experiment, yet in quantum theory, we have at least the potential to verify it with one experiment alone!
Of course, if you're astute enough, you might protest "but we can know the future probability! If I know $x$ and the velocity of the ball I can predict all places it goes and thus reconstruct from present information that yes, '100%' it has that distribution for any future sampling". Alas, though, that doesn't work: with a suitably clever setup, such as that used in Bell's Theorem, you can find that while such a thing makes sense conceptually, the precise statistics it comes up for for all such processes in aggregate just don't match the quantum ones, no matter how cleverly you imagine whatever is going on "behind the scenes". And when we do the experiment on real life systems, the quantum statistics win out. Every last time.
Instead, it seems as if $\psi_x$ somehow represents, itself the actual information that is present in reality, as well as your knowledge thereof, and not just one or the other alone, under the hypothesis of the existence of an incompatible variable.
