Is there any difference between $P$ and $\Pr$ to represent probabilities? I have come across both $P(\dots)$ and $\Pr(\dots)$ being used to represent probabilities. Is there any difference in the meaning of these notations, or are they just different shorthands?

I seem to come by $\Pr(\dots)$ more often in Bayesian probability contexts, though I wouldn't say that's a rule.
They are just different conventions. &nbsp; They don't signify any different meaning.

I personally find the $\Pr$ notation most useful when the discussion involves combinatorics. &nbsp; It distinguishes probability somewhat from permutation.  (Unless you use ${^n{\rm P}_r}$  ...)

It also has that convenient LaTeX command \Pr which renders it in times roman font, and with some space padding, which helps it stand out in a line of multiplied probabilities using just a few keystrokes.
In classical approach to probability defined by Laplace, probability is the ratio of number of favourable cases to the total number of equally likely cases. Here probability of occurrence of A is denoted by P(A). Symbolically, if an event A can happen in 'a' ways out to total 'n' ways then the probability of occurrence of the event is denoted by Pr(A)=a/n