The main difference between nvd and nix-diff is not the output format, I think: it is that nvd works on store paths and nix-diff works on derivations. This makes it natural for nvd to answer the question “what user-visible things changed” and for nix-diff to answer “what causes this derivation to be different”.
On the other hand, I seem to remember the mechanisms are similar for both closures and derivations (in both cases, you call nix-store --query --requisites), so doing both might actually be doable.