Why does it get harder with time, to rotate a screw with a screwdriver? A couple of forces act on the machine screw and create the turning effect. But after sometime it gets harder. So, net torque is decreasing, but why is it decreasing? Frictional force comes into action, but friction doesn't depend on surface area. So, frictional force will remain constant.
So what decreases the net torque of the screw that makes it harder to rotate with time?
 A: When you said that friction does not depend on the surface,well, you learnt this while studying the motion of blocks on a plane. Usually the blocks are frictioning because of their Weight, as such a bigger face exerts small Pressure on the Surface as compared to a smaller one. Since P is W over S, friction stays constant. 
A screw entering a hole is a different case, as every delta L of it brings in its own contribution.
Related: there will be friction screwing at zero gravity, too.
A: A guess: 
1) rust , i.e.oxidation next to the contact area as some air will be there, increasing friction.
2) joining of lattices, as in metalic bonding
That friction increases is evident since solvents (WD40) help loosening stiff nuts .
A: If you run a nut down the length of a rusted screw, the nut progressively scrapes off the rust and gets it jammed into the (very small) clearance space present between the threads on the nut and the screw. once that clearance has been filled with rust particles, the friction force climbs to a very high value- high enough, in fact, to twist the screw apart if you persist in turning it after it locks up. 
as Anna V points out, lubrication will help, but so will unscrewing the nut and clearing off the rust before proceeding further. 
