Quantum mechanics is an abrupt departure from classical models. In Schrödinger's quantum particle mechanics, the function describing particle position $x(t)$ is "replaced" with a statistical object that we call the wavefunction $ \Psi(x,t)$.
Solving Newton's Equation provides direct access to physical quantities that perfectly describe the system over time. In contrast, solutions to Schrödinger's Equation are vectors of probability functions that describe a particle's state, which we must operate on to obtain observable quantities like position and momentum.
To understand how $\Psi(x,t)$ encodes observable information, like particle position, we should start by describing classical motion. In Newton's formulation, a particle's position in three-dimensional space ($\vect{r}$) is described by the differential equation:
$$ \bbox[8px,border:1px solid black] {\vect{F}= \frac{d\vect{p}}{dt} = m \frac{d^2\vect{r}}{dt^2}} \tag{Eq. 1}$$
Eq. 1 - Newton's equation of motion states that acceleration is the second time-derivative of system position. If a system's mass is non-constant, m is not a scalar.
Solutions to (Eq. 1) will describe a system's motion exactly for all time and space; this is a deterministic model for motion. This ordinary differential equation (ODE) leads to the well-known relationships between $x(t)$ $v(t)$ and $a(t)$:
$$ \bbox[8px,border:1px solid black] {a(t) = \frac{d v(t)}{dt} = \frac{d^2 x(t)}{dt^2}}$$Schrödinger's analog to Newton's Equation is a complex-valued energy conservation equation:
$$ \bbox[8px,border:1px solid black]{ i \hbar \frac{\partial{\Psi(x,t)}}{\partial{t}} = - \frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x,t) \Psi(x,t) } \tag{Eq. 2} $$Eq. 2 - Schrödinger's Equation for the energy and motion of a particle. Partial differential equation whose solutions are functions of both position and time: $\Psi(x,t)$.
This is a linear, second-order, partial differential equation, with a similar form to the wave equation. Obtaining analytical solutions to this PDE for a given system potential, $V(x,t)$, is almost never trivial and often it is impossible.
If classical mechanics successfully predicts behavior of many [macroscopic] systems, then those results should fall out of quantum-mechanical models as well. That is, these equations must agree within classical limits of any system that is accurately described by classical models.
The square-amplitude of $\Psi$ is related to $x(t)$ statistically (Gasiorowicz, pp. 46):
$$ \bbox[8px,border:1px solid black]{ \lt x \gt = \int_{- \infty}^{+ \infty} \left[\Psi^* \hat{x}\ \Psi \right]\ dx} \tag{Eq. 3}$$Eq. 3 - Expectation value for position as a function of time. Here, $\Psi^*(x,t)$ denotes the complex conjugate of $\Psi(x,t)$.
In quantum mechanics, $ \hat{x}$ is an operator; $\hat{x}$ represents the operator for position. Operating on $\Psi$ with functional operators produces expectation values of physically significant quantites, such as momentum and position.