How the two routes differ, how many universities, which is easier
There are two ways to apply for GKS and you may choose only ONE. Embassy Track: documents go to the Korean embassy in your country; the embassy screens, then NIIED, then the universities. University Track: documents go directly to one Korean university; the university screens, NIIED approves. Applying through both at once disqualifies you automatically.
Embassy Track: up to 3 universities (for both GKS-G and GKS-U), at least one of which must be a "Type B" (regional, outside the capital area) university. University Track: 1 university only. The A/B distinction also applies to GKS-U in the University Track — Type B (regional) universities receive more quota and see less competition.
In the Embassy Track you compete with your fellow citizens for a country quota; Uzbekistan is among the countries with the most GKS applicants, so competition is high, but the quota is large too. In the University Track you compete with candidates from the whole world for a handful of places at one university, but a strong study plan and prior contact with a professor (for GKS-G) can be decisive. Rule of thumb: an average GPA and a wish for three options — embassy; one specific university and professor (especially master's or PhD) — University Track.
Embassy Track — 3 stages: (1) the embassy screens through documents and an interview, (2) NIIED reviews, (3) your 3 chosen universities accept or reject (at least one must accept). University Track — 2 stages: (1) the university screens, (2) NIIED approves. In the Embassy Track universities can still reject at stage 3 — so make the three choices realistic; do not list SNU, KAIST and Yonsei all at once.