There isn't any kind of legal reason for the distinction; it's just a matter of custom and convenience.
The normal rule is that you would shorthand a case name by referring to the first-named party, but there are lots of times where that is not the case. With Casey, the issue is that Planned Parenthood is a serial litigant, so if you were to just refer to "Planned Parenthood," there would be some ambiguity as to whether you were referring to Planned Parenthood v. Danforth, 428 U.S. 52 (1976), or Planned Parenthood of Southeastern Pa. v. Casey, 505 U.S. 833 (1992), or any of the many other cases Planned Parenthood has litigated in the circuit and district courts.
For the same reason, criminal cases are typically named after the defendant, as we would otherwise have tens of thousands of cases named "California" or "New York" or "United States."