Use the meanings of the prefixes sur- and dis- to explain the difference between surmounted and dismounted.

The prefix 'dis' means apart, away; that is to move away from something while the prefix 'sur' means over, above.
To dismount means to get off something or to remove a thing from its support. For example: The king dismounted the horse majestically.
To surmount means to overcome a difficulty or an obstacle. For example: 
All the political differences between the two countries were surmounted.