Schrödinger's equation is not an equation describing some probability density. It is a fully deterministic equation that takes for input a quantum state |$\Psi$> and tells you how this state evolves with time. If you know the state |$\Psi (t)$> at a certain time $t$ and the Hamiltonian of the system, then you know exactly (at least formally) what the state |$\Psi (t')$> will be at a time $t' > t$.
The fact that the quantity <$\Psi$| $\Psi$> describes the probability density to find a particle at some point $x$ in real space (or equivalently at some impulsion $p$ in reciprocal space, depending on the representation you're using) tells you a different story. It is linked to the fact that the most information you can know in the quantum framework is the wavefunction |$\Psi$>, contrary to classical mechanics where you can track the trajectory of a particle (thus know its impulsion $p$ and position $x$ at every moment).
The "uncertainty" you have in quantum mechanics does not mean we lack some information on the system or that the evolution is non-deterministic. It just means that quantum determinism is different than classical determinism.
Edit : to awnser your comment, yes, it tells you exactly what the system will be at time $t'$. This "exactly" means that you know what the quantum state |$\Psi$> will be at time $t'$, given that you knew what it was at time $t$. It doesn't mean that you know what is the exact position and impulsion at this time $t'$ : this is forbidden by the laws of quantum mechanics (namely the uncertainty principle). The most you can know is the wavefunction. Don't worry, all of this will become clearer as you make progress in your study of quantum mechanics !