Used to refer to groups of people or things, they mean pretty much the same thing, but "remainder" sounds fancier. You could say, "Some of the children stayed home sick, but the rest went to school," or you could say, "Some of the children stayed home sick, but the remainder went to school." In math, a remainder is the amount left over after you do division. So, you'd say, "25 divided by 4 equals 6 remainder 1." The verb "to rest," and its corresponding noun "rest," can't be replaced with "remainder." So, you can say, "This afternoon, I took a rest on my couch," and you can't use "remainder" there.