When you give charge to any sphere like above, charge first spread throughout the conductor and not only surface. A current flows inside conductor moving electrons and changing electric field in conductor.
Now, positive charge then find that outer surface has less potential than inner. They start moving towards surface in form of current. This current stops when all the charges have reached surface to make potential same throughout the sphere, so that charges find no where to move to.
What did you see?
You saw that current moved throughout inside of sphere to reach at surface. Charges were concentrated at surface only when they stopped moving. As a result, electric field at any point inside conductor nullifies.
So, electrostatic is an equilibrium phenomenon.
What happens in wire?
Inside wire you apply an electric field. From positive end charges move throughout conductor to reach negative end. There is no need for them to move towards surface. Electric potential is uniform throughout cross-section of wire. Current is always flowing from one end to other end of wire. Not from center to surface.
Here , you are not letting an equilibrium situation for charge to be concentrated anywhere (not even at ends). This dynamic process hence continues.