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. 
 A: They are just different conventions.   They don't signify any different meaning.
I personally find the $\Pr$ notation most useful when the discussion involves combinatorics.   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.
A: They are just different notation.  Some authors even use the blackboard bold font: $\mathbb{P}$.  What matters is what's inside of the subsequent parentheses (or sometimes brackets, [].)
Several notation species exist for expectation ($E, \text{E},\mathbb{E}$) and variance ($V, \text{V},Var, \mathbb{V}$) too, but they all have the same definition.
A: In probability theory, a typical set-up involves just one probability space $(\Omega, \mathrm P)$ and many random variables $X$, $Y$. Each of them has a distribution $P_X$, $P_Y$ defined in terms of the underlying probability measure $\mathrm P$, for example $P_X(A) = \mathrm P(X \in A)$. In this context, $\mathrm P(E)$ is the probability measure which is anchored to the real world notion of "probability" and events $E$. For example, if $X$ is a random variable modeling a 6-sided die, then $\mathrm P(X = 6) = 1/6$ corresponds really to "the probability of the event that my die is showing a 6 is 1/6". The notation $\operatorname{Pr}$ typically refers to that real-world probability.
