Please don't overtighten the set screw because you're only damaging the copper pipe it slips over.
Typically, when someone installs a tub & shower faucet, they'll install it with plenty of copper pipe sticking out for the tub spout to slip onto. Then, after they finish the wall ceramic tiling or whatever, they'll cut that copper pipe to the required length. When they cut the copper pipe for the spout using a pipe cutter, the pipe cutter leaves a "ridge" at the end of the pipe which needs to be filed off. Otherwise it'll cut up the O-ring of any spout you slip over that pipe.
It could also be any vertigris that's formed on your old pipe as you thought.
Take a small file and bevel the outer circumference of the end of the pipe to remove any ridge left behind by cutting the pipe to length.
Then, take a sand cloth and clean up the outer diameter of that pipe.
Wipe some dish washing detergent on the OD of the copper pipe and the ID of the o-ring inside the spout to act as a lubricant so that you can slide the spout onto the pipe easily.
Just tighten that set screw snug, but not tight. Over tightening does no good, it only does harm.
Finally, read my post in the "Shower in a Tower Block" thread in this forum to learn about a better slip-on spout than you're probably using now.