Is quantum mechanics stochastic or probabilistic?
One works with probability distributions to model mathematically observations and measurements.
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).2
Examples of random phenomena include the weather condition in a future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.
A stochastic process
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.
In classical physics probability distributions are used in experiments where a random process is modeled. Classical physics is deterministic, and various functions have been found to fit particular experimental observations.
It is the randomness for the observations fitted with probability distributions in classical physics that defines the distributions, given the underlying constraints of the individual particle interactions, that can be called stochastic.
The theory of quantum mechanics is probabilistic, the probability is in the axiomatic postulates that are used to model the data at the microcosm of atoms/molecules etc.
The solutions of the quantum mechanical equations do not determine individual trajectories at the particle level, but probability distributions for accumulated data with the same boundary conditions.
An intuitive understanding can obtained by the double slit single electron at a time experiment. The footprint of a single electron looks random, but the accumulation shows interference effect, which can be completely modeled with the quantum mechanical theory. The wavefunction $Ψ$ predicts the probability distribution $Ψ^*Ψ$. That is what is meant when calling quantum mechanics probabilistic. The classically expected randomness can only be described by a probability distribution that differs form the classical probability distribution expected from classical balls.
Bohmian mechanics is not able at present to describe the experiments in high energy particle physics, but there are a number of theorists working on finding an underlying deterministic theory that could give the same predictions as the existing standard model of particle physics. See for example the contributions of G. 't Hooft on this site. (example)