Is there any difference between $P$ and $\Pr$ to represent probabilities?

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.


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.

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