Destroying gov't buildings

[GoTW Kubee]

Are you supposed to destroy other factions' gov't buildings when you capture their city? What are the pros and cons of doing so? I don't understand this part of the game very well yet :/

[Ryoga84]

Are you supposed to destroy other factions' gov't buildings when you capture their city? What are the pros and cons of doing so? I don't understand this part of the game very well yet :/

The Government Buildings are the the symbol of pacification of the province and therefore how well it is inserted in your state.
That said, you should tear down all the foreign culture's GB when you conquer a city (by the way, this give you a small amount of cash). While there are some malus temporal malus in tearing them down immediately (if they are still intact, after the conquest), there is even a small permanente malus in leaving them (foreign buildign in you land usually increase a little the intolerance of the local people).
Most of all, by the way, it's the fact that tearing them down means you're tearing down your enemies governmental system and replacing it with your own, from the point of view of the roleplay it's a great difference.