A polish and compound are differentiated in the abrasiveness. Polishes range from finishing to medium abrasiveness, whereas most compounds are very abrasive and should be used with the utmost care.
Once you have determined which polish or compound gets rid of most paint imperfections, slowly work your way down to a medium polish and lastly a finishing polish to achieve the highest percentage of correction.
The 'cut' refers to the strength of the abrasive, whereas the 'finish' indicates how clean the product will leave the paint. It is not uncommon for compounds to leave holograms in the paint:
But these are not harmful to your car's paint and simply require polishing with a less abrasive product.