Mountain of course ..because you will have a less area to defend from enemy and more troops for defending that area rather than spread out to the whole castle ( for example castle only has on way to get out of it or go inside )
Catapults can't destroy the mountain like they destroy walls ( towers ) and it's easier for archers to fight against enemy troops on the ground while it's harder for enemy to kill them since they are in a very high place
and you can have secret tunnels in the castles to get out of it easily since it has only one way out
so mountain is much easier to defend than a castle with moats surrounded( also requires less resources to build )
If the castle is up high in the mountains then the benefits of having castle there outweigh the negatives. A high positioned castle has less weak spots, and can be often made to have only 2 accessible paths in which the enemies are forced to attack trough. The defenses such as oil, rocks, ballistas become more effective as the enemies troops cant be spread out. Your archers also have an easier shot as the enemy is marching up. His catapults in real life would also have a bad shot, and if it got too close would be destroyed.
A peak is much harder to assault due to the natural protection a mountain gives.
Much harder to tunnel under walls as its bedrock.
Much harder to assault the walls themselves as there are few access points.
Fewer defenders required to oppose any size invasion force due to the narrow approach and the inability to adequately field your army in battle formation,
would be more like an army approaching in column formation, dense compact lines perfect for archers and ballistae on defence.
I always choose mountain or forest. Mountain has a rock surrounded area which helps you to protect a few important castle buildings, also forest type of village has a tree surrounded area which means one side of your castle is definitely protected by the nature
I would choose to design my castle on a flat area in forest or highland type of village, with some manual moat protection. When you use moat protection, it helps you to defend your castle 1-3 more attacks when you're under attack by so many army at the same time.
I will prefer Mountain because definitely there will be a natural castle defense / protection on two or three sides. So I can concentrate on one side by putting my troops to defend the castle very well.
Hmmm... I can see advantages to both, but I think it's a bit easier to defeat a mountain peak castle via starvation than one placed in swamps. I mean, a castle placed on top of a mountain can't be attacked with catapults. The same with one placed in a swamp. You simply can't transport heavy siege equipment through swamp.
I choose a swamp built castle. Even if the walls get destroyed the enemy troops still have to fill in moats. And I don't know if you tried to walk through mud, but it's extremely hard even without an armor on you. Well, I guess that after a while they will simply step on their fallen comrades, but at that point it becomes a Pyrrhic victory.
Stone Mason " Sire, where do you wish to build your castle"
Lord " Mason where would you recommend I place my castle"
Stone Mason " Well sire you could place it in the lowlands which would make it easier for us serfs to slop at the grass to feed your fat belly, or you could place it on the mountain"
Lord " *Sucks in gut* MY BELLY IS NOT FAT, IT IS MODEL OF PERFECTION! Now you foolish peasant what are the benefits of the mountain castle!"
Stone Mason " Well sire it would be more defensible and limit the number of ways your enemies could attack you. It would however make us peasants work harder to get food and resources to the castle just so you can save your turkey neck! Also, many lives will be lost building on the mountain sire, ........" Lord " ENOUGH WITH YOUR INSOLENCE!" Mason" BUT SIRE!......."
Lord " SERGEANT THROW THIS MASON OFF THE MOUNTAIN! If he wants to be so worried about peasant deaths he can be the first one. Also tell the other masons to get to work on the mountain castle, and if the complain throw them off the mountain until they agree to build my castle!"
Sergeant " Sire, I should remind you we only have a few masons left after the last castle you built! Remember you killed them for not completing the castle before the rat attacked!"
Lord " How dare you question me! *Stabs the sergeant with his sword* HEY YOU CLEAN THIS MESS UP, AND GET THOSE MASONS TO WORK OR I WILL BURN THEIR VILLAGE!"
Peasant Mob " We are tired of your tyranny! You kill our masons you threaten to burn our villages! DIE DIE DIE DIE! *Peasants dismember the Lord*"
Rat " HAHAHA you have killed that weak fool for me! NOW GET TO WORK ON THAT CASTLE OR I WILL BURN YOUR VILLAGE"
Only Jesus are able to go over water. Oh stop...Chuck Norris too. But no one else. So you are best protected by moat around your castle. Every * * Noob are able to climb a mountain.
..because you will have a less area to defend from enemy and more troops for defending that area rather than spread out to the whole castle ( for example castle only has on way to get out of it or go inside ) 
