Quantum physics is an irreducibly statistical theory that purports to describe the dynamics of all physical systems at any level, while classical physics isn't a theory but an outdated, incorrect framework for the dynamics of physical systems that we keep around because it is useful and significantly simpler than quantum physics. Determinism is a philosophical perspective:
"Determinism is the philosophical conception and claim that every physical event and every instance of human cognition, volition, and action is causally determined by a continual, uninterrupted sequence of prior events." (italics in original)
from the introduction to Ciprut, JV. (Ed.). (2008). Indeterminacy: The Mapped, the Navigable, and the Uncharted. MIT Press.
This philosophical perspective arose due to the success of classical physics when it was just "physics", and in particular classical (Newtonian) mechanics:
"In physics, the deterministic view developed along with the experimental approach to research, in the sense that phenomena are reproducible under the same unchanged external conditions, implying that the same cause leads to the same consequences under the same conditions...Isaac Newton was the first to lay down the complete basis of classical mechanics, which at the time was considered to be the origin of all physical phenomena...With the rise and development of classical mechanics the view of determinism developed, with the opinion that all natural laws can be described by dynamical equations, either ordinary differential equations (as, for example, in celestial mechanics) or partial differential equations (as, for example, in the dynamics of fluids). In each case precise knowledge of the initial conditions (all positions and all velocities) completely determines the entire future and entire past of the system. When pushed to its extremum, this view implies complete deterministic evolution of the entire universe, including all its smallest and largest details." (emphases added).
From the entry "determinism" in Scott, A. (Ed.). (2005). Encyclopedia of Nonlinear Science. Routledge.
Determinism is not and was never an empirically tested (or derived) theory or component of science, and what empirical evidence existed that it was a correct philosophical perspective existed in the deterministic nature of classical physics (and basically died with it).
"The world most probably is indeterministic, meaning that there are particular events which lack a sufficient cause. Once we grant that there are such events, and that at least some of them are caused, we then require an account of causation that gives the conditions in which they are to count as caused. This is the problem of indeterministic causality. Providing for indeterministic causality has been a major motivation for the development of probabilistic accounts of causation." (italics in original)
from the introduction to Dowe, P., & Noordhof, P. (Eds.). Cause and Chance: Causation in an Indeterministic World. Routledge.
Insofar as determinism means every event is caused by prior events, special relativity renders determinism obsolete all by itself. "Prior" here loses any meaning, as special relativity tells us simultaneity is relative and time itself is not linear except within a subjective reference frame (technically, special relativity holds that time doesn't exist at all, but whether or not the 4-dimensional description of spacetime is ontological is debated). Also, classical mechanics fails to approximate the macroscopic deterministically because even though models of systems in classical physics are deterministic, how some set of variables determines the values of others in such models often depends upon purely arbitrary decision. Complexity and nonlinearity render impossible the ontological determinacy of classical mechanics and classical physics more generally, because despite the deterministic evolution of a system, in order to derive the determined result we have to choose in advance and in complex cases without justification what is going to determine what (by "without justification" I mean that we could pick other variables and accurately determine the evolution of the same system).
The problem determinism always faced was the relative failure of physics in general to describe living systems. The "laws" of classical mechanics were all about external forces acting on bodies and removed from the beginning the possibility for a system to "self-determine". We now know that living systems are qualitatively (not quantitatively) more complex and self-organized than are even self-organizing non-living systems. We also know that complex system exhibit nonlocal (and hence inherently indeterministic) emergent structures. And of course quantum physics makes determinism impossible. What we don't know is whether weaker forms of epistemic determinism sufficiently approximate something like (a form of) ontological determinism so as to justify claims that emergent processes and structures either have no causal power (i.e., causation can't be top-down) and that any causal nonlinearities are sufficiently localized such that a system's evolution is unchangeable under reasonable enough time-scales.
Perhaps the most important and frequent area in which this is investigated and debated is that of mental causation- the idea that emergent (and possibly non-physical) mental states have causal power such that if I choose to do x, the mental state that caused this is closed to efficient causation (it is a process or function that emerges from the brain but is not reducible to the physics governing neuronal dynamics). Currently, the main reason for thinking this might be correct is the influence of late 18th and 19th century perspectives that were themselves byproducts of a scientific framework we now know is wrong.
