There are a number of ways to get at this difference, and the question is a subtle one (check out 3.2 of @gomes_ha's www.repository.cam.ac.uk/items/3cfd12...). So if spacetime curvature isn't a gauge field like the forces we have quantised, is it clear we should quantise spacetime? 5/

A consequence of diff invariance is that the Hamiltonian density is zero. This is different to what we encounter in gauge theories. The quantum version of this, is the Wheeler-deWitt eqn, which implies that the wavefunction doesn't evolve. 6/