At the risk of simplifying, here’s my take:
I. The Southern Chinese cultural background.
The Southern Chinese (especially the Fujianese and Cantonese) have always had the drive to get rich through their own effort, coupled with a business-like mindset. That’s the necessary first step for any kind of development to take place, and one never reached by some other cultures where people are content with merely screwing around, barely scraping by and just surviving.
Why only limited development happened in China in the past was because every few years or so there would be a turmoil leaving people dead, property destroyed, and economy in tatters. Still, people already had the necessary mentality and were willing to repeatedly keep trying and restarting from scratch even if the situation prevented them from achieving their full potential.
Among these people, the immigrants to Taiwan (as well as the Straits Chinese) were already self-selected, with a bias towards risk takers, somewhat similar to what the first European settlers in America were like (although the risks here were smaller by an order of magnitude). Possibly because of this and some other factors, even in the Qing times (late 19th century), Taiwan was already doing better than most other places in China.
II. The Japanese organizational framework.
With the arrival of the Japanese, the system was rigged from the start towards the exploitation of natural resources and shipping them to Japan without anything but the simplest industrial processing beforehand. Formosa was supposed to remain an agricultural hinterland to the metropolis, with the local populace educated only to the extent that it was beneficial towards the economic growth of Japan: for example, the Japanese-established Taihoku Imperial University, now the NTU, [strike]did not[/strike] admitted few Taiwanese students [strike]at all[/strike]. From the Japanese point of view, the Taiwanese were firmly second-class expendable laborers, so the present nostalgia towards these times in Taiwan is more than anything else a manifestation of the Stockholm syndrome and an expression of the disappointment with what happened later.
Still, at least it was all done efficiently, and without killing the goose that laid the golden eggs. Establishing the colony in Taiwan, the Japanese brought with them the much-needed stability. Even with the rent-seeking monopoly system they had set up, the local people could still prosper reasonably well and even advance economically, as long as they kept working hard (this is in sharp opposition to what was brought upon them in the post-war years). As long as no Japanese interests were involved, the legal and economic system was equally fair to all the Taiwanese, or at least predictable in its workings. This was the growth-conducive environment China could not provide.
What also helped was that Formosa was Japan’s first colony, and through the efficient management of it, the Japanese wanted to firmly entrench their position as one of the advanced nations of the world, hereto an exclusively Western club; thus, they didn’t dare attempt some of the stuff the already-established “advanced” nations had no problem doing in their colonies. Had King Leopold’s tentacles reached Formosa, maybe it’d still be rife with civil war to the present day?
III. The post-war policies.
After the post-war takeover, the Chinese republican government continued to run Taiwan very much like a colony but aiming for the short-term gains only as opposed to the long-term sustainability the Japanese were generally trying to achieve. It also brought to the island all of China’s woes, such as nepotism, corruption and gangsterism, to name a few. As a result, Taiwan was largely stagnant until the late 1960s.
One thing the post-war government did well, if only by accident, was the land reform, which allowed mass land ownership, incentivizing rural development and eventually leading to the reallocation of labor towards the nascent industries. As opposed to virtually anywhere else in the world, the land reform was politically easy here, as the land belonged either to the Japanese, who were expropriated, or to the local elites, who had zero political influence with the newcomers; in fact, it was politically desirable to take away their source of income to deprive them of any residual power they might have retained. Note that only after the Communists took over was there a land reform in China, where the landlords were politically well-connected before 1949.
What happened somewhere throughout the 1960s was a combination of many factors at play. The government started refocusing towards encouraging export-oriented growth, as opposed to the earlier attempts at import-substitution policies, which proved a deep failure in basically any country where they were implemented. There was some deregulation, as well as investment in infrastructure and (vocational) education. The government also managed the currency reasonably well since the introduction of the “new dollar,” although still, the banking-sector credit was available to cronies mostly, while unconnected entrepreneurs needed to turn to the kerb market for capital (the “hui”).
Executive summary
To sum it up, I think it’s not as much about as why it all started the 1960s but rather what stopped Taiwan from developing earlier, given all the seeds were planted long time ago.
The answer to this is, I believe, that in the 1960s (1) there was finally no war for some time, (2) the political system became more or less predictable after all the new arrivals settled the boundaries of their respective turfs, and (3) the opportunities for export trade appeared, while (4) the government did not stand in the way and possibly even helped a little.
Yet another take (looking at it somewhat negatively) would be to say that (1) China could not provide a stable and predictable economic and social environment, (2) Japan was not interested in developing Taiwan “too much,” as it would rather have all the industry itself, (3) the post-war period years were largely wasted due to the turmoil brought by the government evacuated from China, until the dust settled in; and that’s why not much could have been achieved before 1896, 1945, and the 1960s respectively.
Edited November 25, 2014 to rephrase the part about the Taiwanese students at the Japanese-era NTU.