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.