Overcome and surmount are synonyms when they are used to mean “to prevail over”. “He was able to overcome many difficulties in his life” means the same thing as “He was able to surmount many difficulties in his life.” However, I would think that “overcome” is more commonly used than “surmount.”  "Overcome" seems more colloquial, and "surmount", fancier.

However, each word has “secondary” meanings which the other word does not share. For example, “to overcome” can also mean to overwhelm or overpower physically or emotionally: “He died in the fire after he was overcome by smoke, and upon hearing the news, his family was overcome by grief.” The word “surmount” cannot be used in this sentence.

Conversely, “surmount” can mean to be above, or rise above, something else. For example, “The Empire State Building in New York surmounts Manhattan.” You could not use “overcome” in this sense.