Can anybody tell me the difference between a castle and a beseiged castle in the game
It appear you are talking about tiles and game mechanics introduced in The Cathers and the Seige tiles from Cult, Seige and Creativity.
These tiles basically do the following to your cities (not "castles"):
The city tiles (and any pendants within them) only count one point a piece when scoring a complete city (two points if the city contains a cathedral). If the city fails to be completed by the end of the game it is worth 0 points.
In regards to scoring of farms, each city is considered to be scored twice at the end of the game if supplied by your farmer (thus 6 points per city or 8 points per city if the farmer also has a pig in the field).
There is also a game mechanic which allows your Knights to "escape" the bbeseiged cities if they border a cloister.
A beseiged city is a city which contains a Cather or Seige tile.