There is not even a single equation in this paper! Not much that I should need to help you understand ...
The neoclassical framework that we have studied assumes that capital is homogeneous and technology is "disembodied"—an improvement in technology (A) increases the productivity of new and old machines equally. Abramovitz describes a world in which new technologies are embodied in new capital. This creates a distinction between capital of newer (more productive) and older (less productive) vintages. [A model with embodied technological progress is explored in Romer's Problem 1.12 and in the papers he cites in that problem.]
The productivity levels in Tables 1 and 2 are labor productivity (Y/L), not "total-factor productivity" (the relationship between Y and an index of all inputs). Capital deepening of the neoclassical kind (increases in k) would increase labor productivity even if A were unchanged.
Questions for analysis
Neoclassical growth models have strong convergence results based on accumulation of capital by countries that are initially below their steady-state capital/labor ratios. On a theoretical level, how are the mechanisms of Abramovitz's "catch-up hypothesis" similar to and different from neoclassical convergence?
On an observational level, how are the outcomes of the catching-up process discussed by Abramovitz similar to and different from the outcomes we would observe from neoclassical, capital-deepening convergence?
What does Abramovitz mean by "social capability" and why is it important for growth?
At the time that Abramovitz wrote, both China and India were poor, slow-growth economies. Since then, they have (to differing degrees) "taken off." Have the social capabilities of these economies changed since the 1980s and, if so, how have these changes been related to their growth experiences?
How do the characteristics that Abramovitz discusses as being important for catching up and/or forging ahead vs. falling behind relate to the "additional variables" that Barro and Sala-i-Martin used in their conditional-convergence regression using the 98-country sample? What other variables does Abramovitz's paper suggest might be appropriate for such regressions? [Just so you don't have to look them up, their variables were "primary and secondary school enrollment rates in 1960, the average ratio of government consumption expenditure ... to GDP from 1970 to 1985, the average number of revolutions and coups per year from 1960 to 1985, the average number of political assassinations per capita per year from 1960 to 1985, and the average deviation from unity of the Summers-Heston (1988) purchasing power parity ratio for investment in 1960."]