What exactly is deterministic in Schrödinger's equation?

The fact is that there are two kind of things: 1) the wave function and 2) the physical observables.

The evolution of the wave function is dictated by the Schrödinger equation and is deterministic meaning that if you know the wave function at some time, then you know it at any time just using Schrödinger equation $$ i \hbar \frac{d}{dt} \left \lvert \Psi (t) \right \rangle = H(t) \left \lvert \Psi (t) \right \rangle \, .$$

On the other hand any observable (e.g. position) is not deterministic in the sense that if you know observables at some time, in general you cannot say anything of the particle at some future time. There exist no equation for observables themselves.

Coming back to the wave function, I have said that it is deterministic. This is true unless someone measures an observable. In that case the wave function collapses in a non deterministic way. Next to the measurement the evolution becomes deterministic again, but just at the instant of the measurement it does not.

In quantum mechanics, the solution of the equations (Schrodinger, Dirac...), called wave functions are deterministic, at each $\left(x,\,y,\,z,\,t\right)$ point, but the only prediction they give is a probability distribution, which depends on the boundary conditions of the problem. $Ψ$ is a complex valued function, and measurements are real numbers and this is the distribution of $Ψ^*Ψ$ (the absolute square or squared norm of $ψ$) which gives the probability of finding a particle at a given space time point.

Probabilities by definition means that many measurements in the same boundary conditions have to be carried out, and a comparison made between the predicted probability distribution and the measured one. So even though the distribution is strictly deterministic, its comparison with one datum is probabilistic.

Note the same boundary conditions statement. Once a measurement is carried out, the boundary conditions are different, a different $Ψ$ is needed for the system after the measurement, which is called a "collapse of the wavefunction". In experiments one does not observe the same particle scattering or decaying, but a large number of same boundary condition set ups to accumulate the probability distribution to compare.

Edit after comment:

By boundary conditions I mean the real numbers that have to be introduced so that the mathematical formula will give predictions for the specific observables of the experiment. For example the energy and momentum for getting the cross section from a scattering experiment of two protons, as in the LHC.

In the standard interpretation of quantum mechanics the time-evolution of the system and what we observe are separated (unlike Newtonian mechanics). The system, while unobserved exists in a superposition of states (all the states that satisfy the Schrödinger equation). These states (while unobserved) evolve in time according to the Schrödinger equation. This part—all the way up to the point of observation—is completely deterministic.

If no observer comes along, then that is the end of the story. If, however, someone comes along and makes an observation, then the system, which prior to observation was in multiple states, is thrown into a single state, and only a single state is thus measured. How that state is chosen is posited by quantum mechanics to be completely non-deterministic.

Note that

1. This interpretation of quantum mechanics is known as the Copenhagen interpretation and is by far the most common interpretation of quantum mechanics and is what’s in all the text books.
2. This general problem is known as ‘the measurement problem’ and is by far the most controversial aspect of quantum mechanics
3. Attempts have been made to add additional variables to the formalism to make the outcome of the measurement deterministic (see Einstein’s EPR paper, for example), however, these attempts cause inconsistencies in the theory of quantum mechanics. Therefor, the current view is that quantum mechanics is a complete theory—I.e we can’t add any more to it whitout generating contradictions.

It’s a very controversial part of the theory of quantum mechanics, but appears to be something we have to live with.

I hope this expiation helps.