Laplace wrote (Essai philosophique sur les probabilités, see also this review)
We must envisage the present state of the universe as the effect of
its previous state, and as the cause of the state to come. An
intelligence which for a given instant would know all the forces by
which nature is animated and the respective locations of the beings
which compose it, if moreover it were vast enough to submit these data
to analysis, it would embrace in the same formula the motions of the
largest bodies of the universe and those of the lightest atom: nothing
would be uncertain for this intelligence, and the future like the past
would be present to its eyes.
According to the mathematician R. Thom, in modern language Laplace's statement is basically just the theorem of existence and uniqueness for the differential equations governing physics (in particular, Newton's second law). Here, uniqueness is important (the future is uniquely determined).
At the time when Laplace wrote the Essai, numerous differential equations were known, but no theorem existed yet stating that their integration was possible for a fairly 'regular' problem. Such a theorem would not be established until some thirty years later (the Cauchy-Lipschitz existence and uniqueness theorem).
Now, it is clear that we're talking about classical physics: to integrate Newton's equations of motion (for any given system, possibly the whole universe), we need to specify the initial positions and velocities of all the particles. This is not a big deal in Laplace's view, as he explicitly speaks of a "supernatural intelligence": his point is that in principle the universe is deterministic.
Issue #1: in practice, the universe is not deterministic for two separate reasons: we do not know how to set the initial condition, and we do not know how to integrate the incredibly complex system of equations (provided that you believe that the theory you are using is the correct representation of nature, ofc).
Issue #2: does quantum mechanics challenge Laplace's view? Time evolution is given by the Schrödinger equation, which is perfectly deterministic: for a given initial state $|\psi(t=0)\rangle$ the future $|\psi(t)\rangle$ is determined uniquely. Of course, the two points in #1 still apply. First, we do not know $|\psi(t=0)\rangle$, we always make a - sometimes physically motivated - guess. Secondly, we do not know how to solve the Schrödinger equation unless the system is quite small and for short times. A possible way for quantum mechanics to challenge Laplace's view is via the measurement problem: according to the standard textbook interpretation, measurement is not described by the Schrödinger equation (it is through the Born rule that probability enters into the theory).
Complex systems are often chaotic: deterministic chaos means that small departures in the initial conditions produce large departures in the following trajectories (this is, e.g., why weather forecast is difficult). This is a practical limitation, I think Laplace was well aware of this on an intuitive level (chaos theory is rather recent).
Following Thom's mathematical definition of determinism, we may say that the uncertainty principle has nothing to do with determinism: it is a mathematical statement on the variance of non-commuting observables (or, even outside the realm of quantum mechanics, quantities related by Fourier transform, see also this video). Statements on determinism are statements on the time-evolution laws of a theory$^1$. Yes, to solve Newton's law you need the initial positions and velocities with infinite precision, but in a quantum world, you need the full information that uniquely defines a state $|\psi(t=0)\rangle$.
The measurement problem is crucial, as it's the possible source of "true" non-determinism: see e.g. On a measurement level, is quantum mechanics a deterministic theory or a probability theory? and links therein (including Is there consensus among physicists that reality is fundamentally deterministic? and Is the universe fundamentally deterministic?). Another, closely related, question is Why was quantum mechanics regarded as a non-deterministic theory?.
Finally, a short quote from the book mentioned in the original question (Brief Answers to the Big Questions, chapter 4):
The wave function contains all that one can know of the particle, both
its position and its speed. If you know the wave function at one time,
then its values at other times are determined by what is called the
Schrödinger equation. Thus one still has a kind of determinism, but it
is not the sort that Laplace envisaged. Instead of being able to
predict the positions and speeds of particles, all we can predict is
the wave function.
$^1$ For example, Langevin dynamics is non-deterministic (as it is based on stochastic differential equations rather than ordinary or partial dofferential equations). However, from a physical point of view, this example does not challenge Laplace's determinism as it is an effective, incomplete, description of a dynamical process (e.g. Brownian motion).