'To conquer' cares little about the destruction caused. 'To overcome' requires summoning psychological strength when in doubt and emotional endurance to continue as one's feelings are lacking in confidence. Psychological conviction helps establish confidence when one feels winning is against all the odds.
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louiST'to conquer' and 'to overcome'The most usual use of 'conquer' is with nations at war. Rome conquered almost all the peoples of the ancient world.