Why was March 4th the original Inauguration Day and why is January 20th the current Inauguration Day, when it is not mentioned in the 20th amendment?
March 4th is the date that Congress first sat under the current constitution:
All subsequent (regular) inaugurations from 1793 until 1933, were held on March 4, the day of the year on which the federal government began operations under the U.S. Constitution in 1789.
January 20th is in the 20th amendment, this was chosen to limit the lame duck session since travel times were less of an issue than in the 18th century. This also limited the incumbent official from taking actions that the people clearly were opposed to during this period as the incumbent technically now had no need to be answerable to the voters (more true when the incumbent was voted out of office rather than chose not to run for another term or was disqualified due to term limits):
Amendment XX Section 1.
The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.