I saw that people sometimes use notation $p(x)$ and sometimes use $\pi(x)$ to denote pdf of a distribution in literature. For example, on page 2 of this paper by Green and Hastie.
Is there any reason to prefer one over another? Like one is more often used to denote posterior pdf or pdf known to a proportionality.
People also use $f(x)$, $g(x)$, or other of Greek and Latin letters... Authors are free to use whatever notation they want. For example, Christian Robert in his book Bayesian Choice writes Bayes theorem using the $\pi$ notation
$$ \pi(\theta|x) \propto f(x|\theta)\, \pi(\theta) $$
What, for me, is nice about such usage of the notation, is that it uses different symbols for "empirical" probabilities and the subjective, Bayesian, probabilities. Moreover, it nicely highlights the fact that posterior is just an updated prior, they are the same species. It is a nice example where special usage of notation improves readability of the formulas.
Basically, if some author wants to use different notation for different kinds of things, then the notation should be defined in their text. There is no laws that impose any use of notation (if there were, then $\pi$ would be reserved for the number $3.1415...$).
$p(x)$, $f(x)$ are common notations for probability densities or probability mass functions, $\pi(x)$ is sometimes used for priors, $m(x)$ is sometimes used for marginal probabilities, but you can find a number of other notations.
