Why do developing countries adopt 21st century technology, but stay behind in output per capita? The economic growth rate of Latin America has been about 3% per year for decades; but if you have ever visited, many of the natives have: internet, smartphones, tomography, hybrid cars, etc. 
Why don't they catch up when they adopt cutting edge technology?
Probably if you go to almost any country in the world, however poor, you will find considerable use of relatively cheap cutting-edge consumer technology such as smartphones and some more expensive cutting-edge technologies used in manufacturing or service industries.  However, there is a big gap between 'adopting' cutting-edge technology at that level and the much more comprehensive adoption of such technology by developed countries.  The benefits of technology in developing countries may be limited by any of the following:


Lack of adequate supporting infrastructure, eg electricity supply
unreliable or subject to regular power cuts, unreliable internet service.
Lack of adequate skills to maintain and make best use of technology.
Limited adoption, eg of smartphones, so that network benefits are
not fully realized.
Lack of capital equipment, knowledge and skills to produce
cutting-edge technology equipment, so that all such technology has to be
imported using scarce foreign exchange; or, for slightly more
developed countries, production of high-technology goods is limited
to low-level assembly work within an international supply chain
where most of the value is added elsewhere.


Suppose, then, that (recognizing the above problems), a developing country tries to adopt cutting-edge technology in a comprehensive way, so as to quickly catch up with developed countries.  Such an approach is heavily criticized in Justin Yifu Lin's book The Quest for Prosperity, which argues, in outline, that developing countries should target industries in which they have a comparative advantage (or can achieve one via improved infrastructure) and which offer an opportunity for gradual upgrading of technology.  While countries differ, this will often mean labour-intensive manufacturing, and it is suggested that industries to be targeted should be those currently dominated by countries whose GNP per capita is about twice theirs.  By contrast, attempting to compete immediately with countries with the most advanced technology is likely to lead to a continuing need for subsidy, distortion of resource allocation, financial repression to channel scarce capital to the subsidised sector, and political problems in handling those profiting from or dependent for employment on subsidies.