Is there only one formula for the statistical difference between a pair of distribution ensembles?
What we call "statistical distance" in cryptography is called total variation distance by statisticians. So it certainly exists outside of cryptography. 

I can't speak to its applications within statistics. But it certainly is the most natural metric for cryptography because it has an equivalent formulation in terms of distinguishing two source distributions given a sample, which is how we state many security properties. Restricting the distinguisher to be polynomial-time leads to natural and useful generalizations.