Cut = wet sanding with 1000-2000 grit sandpaper to knock down orange peel.
Buff = machine polishing with progressively finer compounds and pads.
I did my own cut and buff; it was really scary taking sandpaper to my gloss black paint, yet surprisingly easy (even though paint had cured for 6 months). If you error on the side of caution (ie always start with finer paper and less aggressive compound, and keep away from edges so you don't sand through) it's a time-consuming but rewarding task that will make a world of difference in your paint job. The benefit to having your painter do it is, of course, if he screws up the repaint is on him.
Last edited by Jay Zed; 02/18/2017 10:12 PM.