Question

To keep the current flowing in a conductor continuously we need to maintain a constant ___ difference across its terminal.

Answer 1

To maintain a current you need to maintain the electric field inside a conductor. Only then will there be a reason for electrons to move in the forward direction.
Now to do that one needs to maintain a potential difference across the conductor.
We know that $E=\frac{dv}{dr}$ so if dv =0 (i.e. potential difference is zero) there can’t be an electric field and hence no current.
To maintain this electric field, one needs to ‘maintain’ the potential difference.