The thing that always gets me when looking at Korean history is how the Joseon lasts like 500 years, a ridiculous amount of time for one dynasty. Obviously it wasn't a static entity over all those years, but how much can we say the very late Joseon court resembled the earlier Joseon courts in structure, style, etc?
First off, there was a movement in the 1860s and 1870s led by the Heungseon Daewongun to revert Korea back to its earlier state, so this post applies only to the decades before 1863.
Since you mentioned the court I'll start with the major shift in the structure of the government.
In 15th-century Joseon government the State Council was the most important organ of government, led by the highest-ranking bureaucrats of the court, and the Six Ministries were subservient to it. Military affairs were dealt with by the State Council and the Ministry of Defense. This began to change somewhat in the late 14th century with the formation of the Ministers of Border Affairs (知邊事宰相), an informal council of high-ranking bureaucrats and generals, to deal with the frequent Wokou raids and Jurchen incursions.
In 1510, with riots by Japanese in three Korean ports, the meeting of the Ministers of Border Affairs was reorganized and renamed the Border Defense Council, although it remained a temporary office to be called only in times of external raiding or invasion. By 1555, with a major Wokou attack, the Border Defense Council finally became an official organ of government.
The Border Defense Council soared in importance during the 1590s Japanese invasions of Korea, when it came to deal with a diverse array of duties from appointing governors to organizing religious rites. The Council retained its prominence even after the death of Toyotomi and the retreat of the Japanese, and by the Injo reign the Council of Border Affairs was the most important organ of Joseon government. The State Council became irrelevant. This is highlighted well in a memorial in November 16, 1654
In matters of personnel and rites and music, in matters of finance and the military, in matters of justice and construction, the State Council and the officials of the respective Ministry discussed together prior to taking the decision. The Council supervised over all the matters of the Ministries, the Ministries fulfilled their duties properly, and for a century there were no issues.
In the times of Seongjong the Border Defense Council was temporarily created in the incursion of the Jianzhou Jurchens, and the ministers within it were called the Ministers of Border Affairs.^2 But it was made for the occasional war, and not to deal with all the important affairs of the state.
Yet today [...] the State Council is nothing but a name and the Ministries have lost all their duties. The stated purpose of the Border Defense Council is to "supervise the defense of the borders." And yet even the handling of civil service examinations and the selection of queens and concubines are all dealt by it. [...] In my humble opinion [...] it would be best to abolish the Border Defense Council.
But the Border Defence Council was not abolished. If anything it grew in power in the meantime. And it would remain the center of Joseon government until 1864.
The Border Defense Council ties into the decline in royal power. 15th-century Korean kings, with some minor exceptions, were all fairly powerful with Yeonsangun easily being the apex of royal power in Joseon history.
In the 19th century it's a different case. After purges of another faction that culminated around 1807 the court was dominated by the Andong Kim clan and other families tied to the royal lineage by marriage. By the mid-19th century Korea was completely ruled by the Andong Kims, not the ineffective king Cheoljong. Look at some of the people who were chosen for government offices from just the first three months of the year 1857, with the fact that "Byeong" was the naming scheme for this generation of Andong Kims in mind:
Some other high-ranking government figures of the Cheoljong reign include Kim Byeongji, Kim Byeongdeok, Kim Byeongyun, Kin Byeongjik, Kin Byeongpil, Kim Byeongjip, Kim Byeonghak, and Kim Byeongshi.
The Andong Kims were horribly corrupt, and this led to a number of issues outside the court culminating in a large peasant revolt in 1862, and arguably to the much larger and organized revolt (with religious elements) in 1894.
There are of course a lot more changes than the two I mentioned, both within the court and in Korea in general. Fashion changed a lot. One notable example you'll notice in the court is that the samo or minister's hats were originally this but it had long since shifted to this by the 19th century. Cuisine changed with the Columbian exchange - nowadays who could imagine Korean food without pepper? That made an impact in the court as well.
^2 Not strictly correct. The "Border Defense Council" name is from the early 15th century after the death of Seongjong
