The fundamental difference is this:
One is a STYLE, the other is an OPERATION
Sure one might go into ones surface style and set the slope and or default elevation ranges and the display properties for same. However, and here is the dividing difference, from with the surface style dialogs one can not EDIT the Ranges Values and colors one gets to be something useful, nor can one Output the Analysis as a Legend.
I get your confusion, however keep in mind that IF you have these display components OFF and you perform the Analysis operation you will see nothing, however IF you turn these display components ON what you are seeing will be MEANINGLESS until you perform the analysis function.