What is the difference between St Louis ribs, Memphis ribs, and Kansas City ribs?
There may be some local namings, but normally a St. Louis trim rib is a spare rib with the flap removed (tougher meat) and trimmed to a more uniform size to get a more even cook, especially over a long slow cook.

KC and Memphis ribs, along with Carolina and other regions, normally would not refer to a trim, rather they would tend to refer to cooking and spicing styles.  KC, Memphis, Carolina, Texas, etc. tend to all be known for regional preferences in preparing the ribs.  This tends to include things like:


Do the ribs tend to be smoked, and if yes, how strong?
Do the ribs tend to be wet (with a sauce) or dry (just a spice rub)?
Does any sauce tend to use brown sugar, molasses, or vinegar?
Does spicing tend more to the hot or sweet side?


You will find variations of all of these in every region, but the tendency towards favoring one style or another tends to be associated with a region.  Those style differences tend to also apply to more than just ribs and will include what meats tend to be more common and how they are served.

For instance, for me at least, Memphis BBQ will tend to mean pork.  Ribs yes, but especially pulled pork.  Typically it would be cooked dry with sauce served on the side or added after cooking.  The sauce would tend to be on the thin side, often with at least some vinegar, though less than Carolina, and with some heat.  KC BBQ would include a lot more chicken and beef, especially brisket, but lots of ribs.  It would use a sweeter, thicker sauce with either brown sugar or molasses, and usually the meat would be at least finished wet.