Why do we use $\psi$ instead of a straightforward probability? What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
 A: One purpose of using $\Psi$ rather than just the probability density is to match observation. Dealing only with probability density isn't sufficient.
Imagine you can send particles through two adjacent slits toward a detector screen. You'll find an interesting pattern that looks like an interference phenomenon is happening. This is called the double slit experiment if you want to take a look. There's no way to explain the pattern that appears based only on the probability densities corresponding to each slit alone. In other words, if slit #1 alone produces a density $\rho_1(x)$ on the screen, and slit #2 alone produces a density $\rho_2(x)$ on the screen, there is no way to combine $\rho_1$ and $\rho_2$ to get the actual density $\rho$ that's observed.
Using the probability amplitude $\Psi$ allows us to predict the actual pattern.
I should mention that, historically, this was not the motivation for introducing $\Psi$, but it's a convenient way to view the need after-the-fact. I believe $\Psi$ first appeared in Schrodinger's equation as a way to get the allowed energies of a bound system. This is an equation in $\Psi$, not $\rho$. It wasn't until after the concept of $\Psi$ that the connection $\rho=|\Psi|^2$ was made. Nonetheless, as user36952 pointed out, we have an equation describing how $\Psi$ develops in time, not how $\rho$ develops.
