Why does the representation of the filter matter? How explain it to people outside signal processing theory?
The short answer is that each for each structure there is a different sensitivity to small perturbations of the coefficients. The locus of poles and zeros change under those perturbations. 

The different forms correspond to different factorizations of the polynomials that correspond to the filters transfer function.

So with infinite precision arithmetic the different forms are exactly equivalent but the factorizations are not equivalent with respect to small perturbations.

Similar ideas are found in numerical linear algebra. A Cholesky factorization is more accurate than the unfactored matrix the factorization corresponds to.  This is why rank one updates are favored in various algorithms