What separates "rich" developing nations from "poor" developed nations?
All country classifications are conventions, which depend on the rule used by the particular institution. There is no hard (ontological) rule on any classification. There is no even consensus on the definition of categories. Advanced/rich/developed/industrialized economies are sometimes used interchangeably, but who is in each of them depend on who is talking. Let see an example:

International Labor Organisation (ILO) 



(I had to cut it because it is too long. See the full list in page 67 of this report).

Citigroup



(Report here).

Conclusion

As you can see, ILO puts Czech Republic, Greece, Argentina, and Chile as developed countries, whereas Citi puts them as emerging countries (Argentina as emerging can be seen in other reports). (ILO even distinguishes between emerging and developing).

This shows that there is no hard-coded rule about classification of countries. It all depends on the institution.