This question can be answered at a number of levels of analysis. Any time you cite reason X as the cause of Y, there's naturally a follow-up question of "what caused X". I'll try to organize a general summary of what factors are considered important in the development of South Korea by categorizing them by level of analysis

Proximate/top level causes: Park Chung-hee, a long-serving dictator, was a strong leader who was determined to transform South Korea into an industrial power. He wisely choose a number of smart economic policies--such as selective import-substitution and export-promotion which fostered the development of 'infant-industries', large-scale infrastructural projects, and rural development programs like saemaeulundong (New village movement)--which successfully nurtured the manufacturing sectors of the Korean economy, allowing the South Korean to eventually eclipse the North Korean economy despite an initial disadvantage in both heavy industry and natural resources. Although Park Chunghee's administration deserves credit for smart and successful economic policies, other countries that tried to follow in Korea's footsteps have often failed. This is because Korea's success was also undoubtedly reliant on further underlying factors: primarily Korea's good 'institutions', and 'developmental state' factors.

General Reading: http://en.wikipedia.org/wiki/Trade_promotion_%28international_trade%29 http://en.wikipedia.org/wiki/New_Community_Movement http://en.wikipedia.org/wiki/Park_Chung-hee

Intermediate causes: The main critical juncture that divides the development paths of N. & S. Korea is the adopt of different basic models of economic organization and the 'institutions' embodied within these models (In this response, I'm using the definition of 'institutions' as it used in authors such as Acemoglu; for clarification you may think of institutions as being structural sources of incentives). Although there is definitely cases of states failing while using capitalist models and socialist states out-performing capitalist states, most economists believe that, everything else equal, markets tend to work generally better than commands and controls. The main reason why is due to the self-coordinating ability of markets and the set of incentives (in most cases) tend to be more conductive of pro-active economic activity. South Korea, despite being a dictatorship, had relatively good economic institutions for reasons which will be discussed.

Additionally, South Korea is the poster boy of 'developmental state' literature. The main characteristics of a developmental state can be summarized as:

1)A strong bureaucracy 2)"Development Elites" whose legitimacy depends on development 3)A weak, subservient civil society (No major ethnic, regional, religious or class conflict, which divides people's loyalties away from the state) 4)Mixture of free-markets and & state intervention/industrial policy

So why does S. Korea have good institutions & the developmental state factors?

Further reading: Why Nations Fail - Daron Acemoglu & James Robinson Institutions, Inequality and Growth: A Review of Theory and Evidence on The Institutional Determinats of Growth and Inequality - Richard Bluhm Colonialism, Inequality, and Long-Run Paths of Development - Stanley L. Engerman, Kenneth L. Sokoloff

Fundamental/Root Causes: Korea's good institutions can be generally traced to a few factors:

1. Korea has a long history of meritocracy-based ideals, rooted in its Confucian social values
2. Korea has no significant extractable resources, which often lead to extractive political institutions (see 'Resource Curse') which inhibit growth.
3. After the Japanese Occupation, the Korean war, and significant and successful Land reform, Korea lacked significant historical economic elites ('old money') to rig the system in their favour (see Phillipines or Latin America experience (Sokoloff) for evidence as to how old-money economic elites inhibit growth). Economic elites (emerging within the Chaebols, the large family-owned firms which eventually came to dominate the Korean economy) had to be 'self-made' elites, engaging in productive activities to become elites rather than engaging in 'rent-seeking' behaviours.

Why does Korea have the factors which characterize a 'developmental state'?

Weak and subservient civil society--Korea is a homogenous country: all Koreans share the same ethnicity, the same Confucius values, and other divides within Korean society (Gyongju vs Jeolla regional rivalries, Christian vs Buddhist divisions) were not severe enough to impede government activities (until the Gwanju uprising, where Jeolla resentment towards Gyongju dominance in Korean politics reached high enough levels to spark major political strife and played a major role in Korea's transition to democracy). Furthermore, in the aftermath of the Japanese Occupation and the Korean War, there was a strong nationalistic feeling of solidarity among Koreans, accompanied with a sense of duty towards doing whatever possible to rebuild Korea.

'Development Elites' legitimized by growth--Due to the Northern threat and western influence, Chunghee's dictatorship relied on rapid economic growth to legitimize itself. If Chunghee's administration failed to produce growth, it would lose legitmacy among its own citizens and among its international allies, who held off pressure for democratization in part due its ability to act as a counter-example/balance to communism.
In contrast, N.Korea's leadership had less legitimacy concerns, as it was able to play Mao's China and Stalin's Soviet Union off each other. And even in regard to its Soviet and China Allies, its support was conditional primary on ideological matters rather than development matter.

State Strength--Korea's historical bureaucracy/Confucius legacy and Chunghee's strong rule also allowed the formation of a strong bureaucracy and a strong, interventionist state.

I won't go through N. Korea's economic history, as it is strikingly less spectacular. The gist of that story is essentially the ideological choice that N. Korea made at the outset led to a situation where bureaucratic elites were allowed to rule based on an ideological-basis and benefit under rent-seeking behavior that inhibits the development of a more inclusive society.

Sources: Of Miracles and Models: The Rise and Decline of the Developmental State in South Korea, John Minns Third World Quarterly Vol. 22, No. 6, The Post-Cold War Predicament (Dec., 2001), pp. 1025-1043 https://www.kdevelopedia.org/Resources/territorial-development/land-reform-korea--04201306110126628.do?fldIds=TP_TER%20TP_TER_NA

Sorry for the sloppy sources. I don't have access to my library at the moment which is relevant to this topic, but if there's any specific issue you want to look more into I can probably dig up additional sources.

EDIT: Apparently I can't figure out how to use bullet point or format properly. Further apologies.

I wrote a comment awhile back that sort of addresses your question:

1. The Sino-Soviet Split. North Korea took a distinctly Stalinist approach to Communism. As Kim Il-sung became more and more of a Stalin-style dictator, the Maoist Chinese began to view him as a revisionist. Thus the two countries stopped being as close as they had been in the 1950s. Instead of aligning more with the Soviets, Kim decided to develop a state that was as independent as possible from the rest of the world. Thus, we get Juche: basically North Korea's way of being separate from the world.

2. South Korea began to do better economically. Under Park Chunghee, South Korea jumped from an agricultural economy to a full blown industrialized nation. With the South outpacing the North, Kim's dream of reuniting Korea under favorable conditions for the North were over. Plus, Park began to create a distinct South Korean identity through rebuilding lots of old palaces and monuments, as well as creating a distinctly pro South historiography.

3. The South began to do better militarily. Again, under Park, the South began to receive updated weapons from America due to him sending about 300,000 troops to Vietnam. So not only was Northern economic domination unlikely, but military domination as well.

So, the North alienated China and the USSR, and South Korea proved able to exceed the North in every way without them. This makes their list of friends pretty thin- who else would help them? As to establishing a personality cult, I understand that Kim Il-sung followed Stalin's method- find a group of people, use them to eliminate your enemies, then kill them all using a new group. Rinse and repeat until you have a full blown personality cult. He did this with great success using a lot of the guerrilla leaders that helped fight off the Japanese in Manchuria. Sources: A Review of Korean History. It details both Koreas during this time. It is VERY pro-Park Chunghee, but also one of my only references for Korean history. Understand that Park was also a fairly very repressive dictator for the South. "Nothing to Envy." Which details the lives of a few North Korean refugees that fled to the South or China.

To make that a bit more relevant: although the DPRK was more developed than the South, it wasn't quite yet a modern country by the end of the Korean War (which the war damage certainly didn't help). The only reason that the ROK managed to have the economic vitality that it did was because of outside help. Since the DPRK didn't have an ally that was a really "hard core" ally after the 1960s, they couldn't develop as well as the South.