As long as no measurements are performed, quantum mechanics is perfectly deterministic. That is what we mean when we say that 'time evolution is unitary'.
The wave function evolves in time according to Schrödinger's equation. Just as in classical physics, if you know it at a given time, you can compute it for any time in the future. The problem with measurement (in the standard, Copenhagen interpretation) is that it involves an external observer. The system evolves deterministically, but for reasons that are exterior to the system, someone performs a measurement. This is a sudden and violent event where the whole wave function of the system interacts with another incredibly complex system. This usually collapses the wave function onto eigenstates of observables in an unpredictable way.
Within the many-world interpretation, there is no external observer. Instead, our view of the system is restricted to one single outcome of the measurement. I would not call this determinism, because according to this interpretation, there is no possible way for us to interact with other 'worlds'. Most of the information necessary to predict the future is hidden from us. Most importantly, this is built into this theory and can not be avoided.
However, as CuriousOne says, the difference in between different interpretations of quantum mechanics is pure philosophy. Whatever the interpretation you subscribe to, the outcome of the calculations that you make will be the same. Choose the interpretation that you prefer, it makes no difference at the experimental level.