Is there a standard rule for the choice of the letter, or, does it have something to do with how the words evolved from the root words to the current words in the specific case? (This is referring to American English versions of the words, although I wouldn't mind the British answer as well)
Defense is AmE while defence is more common in all main English dialects. The origin is from Old French defens (without the final e), which was later assimilated into defense, but not before it inspired the alternative spelling defence, via the same tendency that produced hence (hennis), pence (penies), dunce (Duns).
Evidence is from Old French "evidence" where the suffix was and remained -ce.
Defence and defense are different spellings of the same word. Defense is preferred in American English, and defence is preferred in all other main varieties of English, including Australian, British, and Canadian English. The spelling distinction extends to most derivatives of defence/defense, including defences/defenses and defenceless/defenseless. But the words defensive, defensiveness, and defensively have an s everywhere.
Though defense is now the American spelling, it is not American in origin. The OED and Google Books reveal examples of the spelling from as long ago as the 1300s, many centuries before the United States existed. That spelling continued to appear a fraction of the time through the 19th century, when it was taken up by American writers. Today, to the chagrin of those who dislike American English, the spelling is gaining ground throughout the English-speaking world.