Off the top of my head, a few things:
The biggest reason I can think of is that, with the exception of Brazil, many of the independence wars were bloody affairs and fought primarily if at all, in Latin America. So by the time "independence" was achieved much of the already basic infrastructure (I think I may be wrong in calling in basic, but Im not sure) of these new republics was wrecked.
Without the support of their empires and any type of industry within these countries had to rely on international loans to rebuild which made it harder to develop economic independence.
I put "independence" in quotation marks earlier because the fact is that a lot of these wars were originally fought so that the creole elites could reap the spoils of their colonies which Im pretty sure was the case in Central America, Mexico, the Southern cone countries and the Caribbean.
Point being that the vast majority of the Latin American populous wasn't exactly benefitting from independence, though they were told they would by military leaders in order to have people to fight their wars. This was largely the case for African and Indigenous groups I believe.
Building off the last point, the Creole elite did not always see itself as one entity. So that there was a lot of in fighting within ruling groups as well as regionalisms fueled by both philosophical and economic reasons, which led to border disputes in South America and the fragmentation of the central american republic into smaller states.
A US interventionist policy of the western hemisphere known as the Monroe doctrine which they really only used when it benefitted them, like when they took Panama from Colombia, or when they took over Latin Caribbean policy making after the Spanish American war, or when they protected US business interest in central America.
Later on this mentality would lead to the support of dictators to persecute populist and communist movements during the cold war which further stagnated and held Latin America back, but it was years after the "independence" phase of latin america.
There was in fact a culture of education, centralized around the church and the wealthy, but it existed. Latin america even has some pretty amazing poets, writers, and philosophers participate in the wars for independence i.e. Jose martí in Cuba, and Miguel Hidalgo in Mexico. Though the lack of good education programs was detrimental to Latin America, as the other poster stated a lot of the set backs for some Latin American countries came from US manifest destiny aggression. Such is the case of the Mexican American war, Spanish American war, and the taking of Panama. But the more important obstacles were the post independence regionalisms and the loans that Latin American countries took out, which placed these countries in debt from the get go.
Please keep in mind that when I say Latin American countries took out the loans I am speaking to the creole elite group that ran the country, and usually the money went to maintaining power not to nation building.
I want to emphasize here in case it isn't obvious, if Europe and the US hadn't tried so hard to gain economic and political control of Latin America and had the creole elite who were white Europeans been more trusting and willing to help and work with indigenous, and african groups post independence, Latin America would have had at the very least a fighting chance.
At the end of it all independence signified the start of change but not the end of troubles for Latin America.
Hope that made sense! Read born in blood and fire or latin american politics by Gregory weeks if you want a better understanding, as those are my two sources.
Do you mean technologically, GDP or what? For one, they got their independence and identity much more recently. European-like rate of literacy is a relatively new phenomenon in South America. There were no universities, journals, or a tradition of education, trades and manufacturing. South American countries were not even really countries when they were conquered by Spain or Portugal, for example. It is later through various nationalist movements that they united into what we know today. For example you can google "Gran Colombia" and Simon Bolivar to learn about the formation of some of these countries.
There was also a lot of intervention from the US, which by some is attributed to restraining the natural progress of certain countries (US was preoccupied with stopping communism/socialism around the world and especially in "our back yard" so some Latin American countries had democratically elected leaders overthrown with US backing, and replaced with someone who will play nice with US companies (For example, openly Fascist dictators like Pinochet). Some however defiantly remained in power, like Castro, or more recently Chavez. In case of Castro this lead to sanctions and the embargo, which naturally hurts Cuba's economy to be shunned by one of the world's biggest markets. Basically what I'm trying to say is the US played an important role economically in the development (or lack of) of our southern neighbors during the post-colonial period.