Interesting question. I tend to use them nearly synonymously, though I could apply proselytize to aggressive argument in favor of any point of view, whereas I would always assume evangelize meant preaching religion, and in practice Christianity specifically. I did a bit of looking, and my OED gtraces the etymology and changing usage of the two words. In the New Testament, a “proselyt” was a convert to Judaism, and the word has been used in English with the primary sense of to convert since the 1300s. Beginning in about the 1700s, the connotation of zealotry and aggressiveness, with a negative feeling, began to accompany the use of the word. Today, it seems to generally mean trying to convert by argument to a point of view, usually religious.
Evangelism, and all its derivations, are much more frequent in the New Testament and have the sense primarily of proclaiming the “good news” of Christ’s advent and our subsequent salvation. It was used of those converting the heathen, but it meant primarily preaching or talking about Christianity more than trying to convert. During the reformation, it was applied to “fundamentalist” sects such as Calvinism and Methodism that emphasized the intrinsically sinful nature of man and the insignificance of “works” for salvation. It acquired its negative connotations mostly in the last century (20th) for those of us not interested in the “good news” and bothered by the particularly conservative denominations of Christianity. Today in America, it seems to be applied mostly to politically conservative Christians, though some of the surveys on political attitudes suggest there is a surprisingly large minority of Christians who identify themselves as evangelicals but don’t fit the classic political type.
According to Wikipedia, among evangelical Christians different groups have different criteria for distinguishing prosyltising from evangelising, and they may forbid the first but require the second. Say a missionary is considered to be evangelising if he goes to the Third World and does aid work while talking to the people he is helping about Christianity, but he is forbidden from requiring conversion in exchange for giving aid. I don’t think there is much consistency in how these distinctions are applied from one church to the next. I’d be interested in hearing what others think about the distinction, especially those who have been or are part of an evangelical denomination (didn’t DJ Groethe used to be an evangelical Christian?)