连续 is usually used as an adj./adv. word; while 继续 as a verb.
这几天连续下雨。（地 is omitted, should have been 这几天连续地下雨。） It rains today, yesterday, the day before last, ... (so many continual rainy days)
今天继续下雨。 （昨天 is omitted, should have been 今天继续昨天，下雨。） It rained yesterday, and it continues raining today. (The rain of yesterday didn't stop, and lasted to today.)