What is the difference between "conquered" and "won"?
Conquer has the sense of either taking by force (seize), to put down by force (subdue), or to overcome something difficult (ie. conquer your fears).
Win is to become first in a contest or competition, acquire through effort or skill, obtain an advantage, or attain a goal.
So, there is some overlap in meaning, but difference in emphasis. Consider:
At the start of World War II, Germany won France.
You could say this, if you mean to emphasize the fact that they acquired control of France through superior tactics or strategy. But most people would say:
At the start of World War II, Germany conquered France.
Since most people would emphasize that they seized (most of) France through aggressive and unprovoked military action.
I know this is a bit old, but it seemed worth pointing out that in gaming circles, there's a distinction between receiving a positive game-over screen ("beating" the game, or "winning") and completing 100% of the game (including optional sidequests and collectables); some gamers speak of "conquering" the game when they reach 100% completion, as opposed to merely "winning" the game.