"A week is seven days."
주 means a round/cycle and has been applied to weeks but you have to add words to set something as a duration. 일 is certainly being used in its sense as a counter word in the Korean. One week of days is seven days. In Japanese the equivalent word for 주일 could also be counted as six or five days depending on to what extent the weekend is included so this is informative to me -- in Korean it seems fixed at seven . . .
주일 and 주 are synonymous. (주, poss. abbreviation of 주일)
Both stand for "week(s)" and both can be used as counter for week. 주 is found more in common usage.
일/한 주일 = 일/한 주 = 1 week
이/두 주일 = 이/두 주 = 2 weeks
이번 주일 = 이번 주 = this week
지난 주일 = 지난 주 = last week
다음 주일 = 다음 주 = next week
일주일 내내 (all week long) = 일주 내내 (throughout the week)