Calculating load regulation in this zener diode circuit
Taking the derivative of VL with respect to Vs and handling iL as a constant, I get the result [...]
  However (!), if I'd written the last term in KCL like this: [...] I would get a different result.
  Why? How do I know if I have the correct equation to take the derivative?


The difference is that you cannot assume iL as a constant.
iL changes when Vs changes. So, the assumption iL could be handled as a constant is incorrect.