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This is what I learned from textbook. An insulator is insulate as the gap between the valence band and the conduction band and the fermi level lies in the gap. A superconductor is super electronically conductive because there is a gap between the BCS ground state with the first excited state. This gap prevents electrons from being backscattered. I'm trying to understand why both have gaps but one is insulator, the other is superconductor.

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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.

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    $\begingroup$ Thanks so much. Cooper pair, as far as I know, is made of two electrons that carry opposite momentum and spin. So the net momentum is zero. This is similar to the fully occupied valence band. Can you explain this further? $\endgroup$
    – ruima86
    Commented Jul 8, 2016 at 14:57
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    $\begingroup$ @ruima86: see Why are Cooper pairs formed by electrons of opposite momentum and spin?. The pair is initially formed by electrons with opposite momenta so it initially has zero momentum. But this is just its initial momentum, and when a field is applied the momentum of the pair becomes non-zero leading to a current. $\endgroup$
    – John Rennie
    Commented Jul 8, 2016 at 15:07
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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.

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    $\begingroup$ The BCS ground state consists of cooper pairs that are formed by two electrons of opposite spin and momentum. To me this is quite similar to the fully occupied valence band in which existence of electrons of opposite momentum resulting in no net current. Can you further explain the difference between the ground states? $\endgroup$
    – ruima86
    Commented Jul 8, 2016 at 14:49
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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.

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    $\begingroup$ Hi, thank you for your nice answer. What is the meaning of "one-particle gap" and "two-particle gap", do you have a reference or could you provide a quick explanation? $\endgroup$
    – Quillo
    Commented May 19, 2020 at 10:18
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    $\begingroup$ Sure, did you try looking at arxiv.org/abs/1011.3275? $\endgroup$
    – Tamaghna
    Commented May 20, 2020 at 21:36
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Normally Fermi energy is temperature dependent. We define the Fermi level at absolute zero. An insulator at temperatures near absolute zero have very less energy compared to the conduction energies. A superconductor state holds good at very low temperatures (liquid He temperature). The superconducting state is characterized by cooper-pair, which is not to be treated like a normal conduction electron. A normal conduction electron well satisfies Maxwell's equations, but a superconductor do not. So it is better not to mix the ideas of insulators and superconductors. For illuminating discussion on superconductivity, see Introduction to Solid State Physics by Charles Kittel

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For an insulator, the energy gap is the difference in the energy of the conduction band and valence band. But for superconducting state, "Gap" means the difference between energies of individual electrons in the Cooper pair near the Fermi energy ($E_F$). These Cooper pair electrons are entangle in spin, position, and momentum, etc.

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