The symmetric class-regular (4,4)-nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with G @ \mathbb Z₂ ×\mathbb Z₂G \cong {\mathbb Z}_2 \times {\mathbb Z}_2 , and 13 nets with G @ \mathbb Z₄G \cong {\mathbb Z}_4 . Using a (4,4)-net with full automorphism group of smallest order, the lower bound on the number of pairwise non-isomorphic affine 2-(64,16,5) designs is improved to 21,621,600. The classification of class-regular (4,4)-nets implies the classification of all generalized Hadamard matrices (or difference matrices) of order 16 over a group of order four up to monomial equivalence. The binary linear codes spanned by the incidence matrices of the nets, as well as the quaternary and \mathbb Z₄{\mathbb Z}_4 -codes spanned by the generalized Hadamard matrices are computed and classified. The binary codes include the affine geometry [64,16,16] code spanned by the planes in AG(3,4) and two other inequivalent codes with the same weight distribution.These codes support non-isomorphic affine 2-(64,16,5) designs that have the same 2-rank as the classical affine design in AG(3,4), hence provide counter-examples to Hamada \mathbb F₄{\mathbb F}_4 -codes spanned by generalized Hadamard matrices are self-orthogonal with respect to the Hermitian inner product and yield quantum error-correcting codes, including some codes with optimal parameters.