Question

Why the term 'hidden symmetry' is preferred compared to 'spontaneously broken symmetry'?

Answer 1

I prefer in some ways the term hidden symmetry. The reason why the term hidden comes in is that the symmetry of the vacuum or ground state is restricted away from the symmetry of the Lagrangian. This is also seen in the dichotomy between what is called the false vacuum, which is the high energy vacuum with the symmetries of the Lagrangian, and the physical vacuum of lower energy that has a restricted symmetry.
The spontaneous aspect of this comes with the exact state that emerges in the breaking of symmetry, which is in a Ginsberg-Landau type of theory stochastic and not dynamical. There are theories which posit dynamical symmetry breaking, but they have largely been a "minority report."

Answer 2

The word hidden symmetry gives me the impression that the symmetry is not actually broken.