“Developing countries” is a neoliberal euphemism which implies that poor countries are poor because they just havent developed a proper capitalist infrastructure yet. The phrase “third world” comes from the “Three Worlds Theory” of Mao Zedong which divides nations into categories based on the patterns of exploitation they fall under (with “third world” nations being colonized and exploited).
“Third world” identifies oppressed nations as being colonized and exploited, “developing” implies that they have failed to adopt neoliberal policies and open themselves to foreign capital.