Energy. A time crystal is at its lowest energy state.

A harmonic oscillator has energy in it - if you took away the energy it would stop moving.

i.e. to stop a time crystal add energy. To stop a harmonic oscillator take away energy.

The other responses have not addressed the possibility that you are referring to a quantum harmonic oscillator. If you are, that is a very good question. I believe the resolution here is that while the phase of a quantum harmonic oscillator in the ground state changes with time, the results of bracketing any hermitian operator (i.e. measuring a physical quantity) does not change throughout time. Therefore in the ground state of the quantum harmonic oscillator, the system is time translation invariant. So a quantum harmonic oscillator does not form a time crystal.

Well, yeah, I thought it was a given that we're in quantum-land here. In classical physics the lowest energy state of a system is necessarily static.

I believe that the idea is the time crystal remains in its ground state the entire time (especially given the work of the author on anyons).