What is the difference in meaning between the notation $P(z;d,w)$ and $P(z|d,w)$ which are commonly used in many books and papers?
I believe the origin of this is the likelihood paradigm (though I have not checked the actual historical correctness of the below, it is a reasonable way of understanding how iot came to be).
Let's say in a regression setting, you would have a distribution: p(Y | x, beta) Which means: the distribution of Y if you know (conditional on) the x and beta values.
If you want to estimate the betas, you want to maximize the likelihood: L(beta; y,x) = p(Y | x, beta) Essentially, you are now looking at the expression p(Y | x, beta) as a function of the beta's, but apart from that, there is no difference (for mathematical correct expressions that you can properly derive, this is a necessity --- although in practice noone bothers).
Then, in bayesian settings, the difference between parameters and other variables soon fades, so one started to you use both notations intermixedly.
So, in essence: there is no actual difference: they both indicate the conditional distribution of the thing on the left, conditional on the thing(s) on the right.
$f(x;\theta)$ is the density of the random variable $X$ at the point $x$, with $\theta$ being the parameter of the distribution. $f(x,\theta)$ is the joint density of $X$ and $\Theta$ at the point $(x,\theta)$ and only makes sense if $\Theta$ is a random variable. $f(x|\theta)$ is the conditional distribution of $X$ given $\Theta$, and again, only makes sense if $\Theta$ is a random variable. This will become much clearer when you get further into the book and look at Bayesian analysis.
$f(x;\theta)$ is the same as $f(x|\theta)$, simply meaning that $\theta$ is a fixed parameter and the function $f$ is a function of $x$. $f(x,\Theta)$, OTOH, is an element of a family (or set) of functions, where the elements are indexed by $\Theta$. A subtle distinction, perhaps, but an important one, esp. when it comes time to estimate an unknown parameter $\theta$ on the basis of known data $x$; at that time, $\theta$ varies and $x$ is fixed, resulting in the "likelihood function". Usage of $\mid$ is more common among statisticians, while $;$ among mathematicians.
Although it hasn't always been this way, these days $P(z; d, w)$ is generally used when $d,w$ are not random variables (which isn't to say that they're known, necessarily). $P(z | d, w)$ indicates conditioning on values of $d,w$. Conditioning is an operation on random variables and as such using this notation when $d, w$ aren't random variables is confusing (and tragically common).
As @Nick Sabbe points out $p(y|X, \Theta)$ is a common notation for the sampling distribution of observed data $y$. Some frequentists will use this notation but insist that $\Theta$ isn't a random variable, which is an abuse IMO. But they have no monopoly there; I've seen Bayesians do it too, tacking fixed hyperparameters on at the end of the conditionals.