So what is the difference?
Generally, a Polish is where you use wax in the same colour as the shoe (and some neutral) to maintain a high shine on the shoe in the existing colour of the shoe.
Patina is where we either have a shoe of un-dyed leather or a shoe that has had the colour mostly stripped out of it. Which is then dyed with a variety of colours and then polished to highlight the textures and colours.
Antiquing is where we take a new shoe and use polish to create the effect of wear and age to individuate the shoe to the wearer.