And how do you compare the benefits of vegetables versus fruits?
"Benefit" and "advantage" are synonyms (see list for Words 1-10). Both are used as nouns in the example sentences here and in your workbook, but they can also both be verbs that mean "to be helpful or useful to."
a justification for something existing or happening
Winning the World Cup would be cause for the world's biggest street party.
A cause can also be a goal that people support and work toward. A soccer goal is too small to be considered a cause. But the goal of winning the World Cup and bringing honor to one's country could be considered a worthy cause. If that happens, then that would be a reasonable cause to have a really big party.
Sometimes that can pose the most difficult challenge because you can only advance to a level that your mind perceives is possible.
Note the letters of the word "call" are also in "challenge"--this might be helpful for remembering that you can challenge someone by calling the person out for a contest or fight. Or you can challenge an idea by calling attention to it and demanding an explanation. Both "call" and "challenge" can be used as nouns or verbs.
a trait that sets someone or something apart from others
Well, it was explained, we have to match your eye colour and characteristics.
"Characteristic" is used as a noun here. But it has the suffix -ic which is characteristic of adjectives. As an adjective, it means "typical or distinctive." Eye color is a physical characteristic; members of the same family might share a characteristic trait of violet eyes.