In my experience -
Royal shine is an aggressive polish, that smooths the surface and puts a shine on the ball for more length. (All polishes contain micro-abrasives which smooth the surface.) It should work for any base grit level, from 1000 - 4000 grit. The more you apply, the more you press for longer periods, the higher the gloss and the finer the finish, up to as much as 5500 grit or thereabouts.
Royal compound is not so much a polish but a "compound" with lots of rough aggressive abrasives in it. You can take a 500 grit surfaces to much higher grit levels in stages because the abrasives break down with pressure on the spinner, getting finer and finer. The main problem with compounds vs. abrasive pads, is the final surface is not related to any specific grit level. The final result depends "on your eyeball". In general, you will take it to somewhere between 1500 and 4000 grit.
RC will put a shine on the ball if you apply a lot with a decent amount of time on the spinner OR if you start with a finer base grit level, like 1500 or 2000 grit. But if you do that, you might as well just use Royal Shine.
RC takes more experience and testing to get consistent results. RS is easier to get a shine, but RC can offer a wider range of final surfaces.