The reason for this change is a lack of compatibility between quad-core processors and LTE. That said, there will soon be a Galaxy S III that supports LTE, while still carrying the Exynos processor. To do so, Samsung said that the Korean LTE Galaxy S III "includes a quad-Core Exynos application processor and an LTE modem chip separately."
That explains why that particular LTE version has increased in thickness by 0.4mm, to 9mm. It wasn't an increase in battery capacity. The battery is the same 2,100mAh version found everywhere else.
The Korean version of the GS3 also supports Voice over LTE, which all three major Korean carriers plan to roll out the service towards the end of 2012. Meanwhile, Verizon earlier said that its own Voice over LTE support would not arrive in a national roll-out until early 2013.