"Developing countries" is a much better term. "Third world" is vague, outdated, and means something completely different.
Not that the US is a developing country; it was developed, and is currently degenerating. It's still very rich, though. So instead of trying to catch this in a single ill-fitting word, perhaps it would be better to simply discuss the actual issues: the rise of poverty, lack of social mobility, crumbling infrastructure, etc.
But today it mostly means "developing countries" (which is actually an euphemism for "stuck in underdevelopment")