Since I have the opportunity to play a game of Janggi (Korean chess) tomorrow, I made a summary of the differences in game play to Xiangqi (Chinese chess) for myself. 99% of this is based on this page at chessvariants.org. Here is my summary:

Setup and Start

• Kings start at the center of the palace
• Each player is allowed to flip horse and elephant in position on one or both sides; red first, then green/blue
• Red does not start, blue/green does.

Movement of Pieces

• Rooks / Chariots
  • Can move diagonal within the palace (if the whole move remains a straight line)
• Elephants / Ministers
  • Move like a big horse: one straight, two diagonal. All passed ground needs to be empty.
  • Since there is no river, elephants can travel all the board
• King / General
  • Can move on marked diagonal lines, too
• Advisors / Guards
  • Can move on marked straight lines, too
• Cannons
  • Can move diagonal within the palace (straight line)
  • Need to jump (over a single piece) even when moving without capturing
  • Cannot capture other cannons
  • Cannot jump over (any player's) other cannons
• Soldiers / Pawns
  • Can move sideways right from the start (there is no river to promote them)
  • Can move diagonal within the palace (though never backwards)
• (Horses / Knights behave the very same as in Xiangqi)

Special rules

• Different approach to flying generals / face-to-face laugh
  • Allowed
  • Puts the other player's king in check
  • Revokes your right to win! Can only used in hope for a draw
  • And: If the king is used to defend a piece attacking the other king to checkmate, it is considered a draw (from this page on ancientchess.com)
• Passing (skipping a move) allowed only when
  • you are not in check yet and
  • any move would put you in check.
• Movement in the palace
  • Soldiers, cannons, rooks are allowed to walk marked diagonal lines in the palace (respecting their base rules, e.g. pawns not backwards, only straight lines).

If you spot mistakes, ambiguities or missing things (expect draw-related point counting) I'd be happy to hear from you. Thanks!