What is the difference of the gap between superconductor and insulator?

The difference is that in a normal conductor the current is carried by fermions (i.e. electrons) while in a superconductor the current is carried by bosons (i.e. Cooper pairs).

Have a read through my answer to What is it about the "conduction band" of a material that is distinct from the valence band? where I explain why a full energy band cannot carry a current. In a conventional conductor any momentum eigenstate in the band can be occupied by at most two electrons (with opposite spins) so in a full band the net momentum of the electrons in the band is zero i.e. there is no net drift velocity and hence no current.

In a superconductor the electrons pair up into Cooper pairs that obey Bose-Einstein statistics, so any number of Cooper pairs can occupy the same momentum state. That means the electrons joined into Cooper pairs can have a net momentum, and hence a net drift velocity, so they can carry a current.

Existence of a gap means that ground state and excited states are well separated and a transition from ground state and excited states requires some energy.

Existence of a gap does not determine whether a system is insulating or not. In your case, conductivity is determined by the ground state. For an insulator, the ground state is insulating while ground state of superconductor is superconducting. A gap here means it is not easy for insulator to be excited to carry currents and for superconductors to lose its superconductivity.

In a superconductor, there is a one-particle gap but no two-particle gap. In a true insulator, all n-particle gaps are finite. With a small electric field you can still create gapless two-particle states in a superconductor, which carry a current. This current is non-dissipative because all particles are in a single macroscopic coherent state (the ground state), unlike a metal where the current is carried by single particles in many incoherent single particle states.