From a dictionary lookup, it seems that both 周 and 星期 can mean "a week"—is there any difference in usage?
周 can be a lot of things, while 星期 can only mean a week.
周 has the meaning of a circle or circumference. One week is just like time running in a circle in 7 stages or 7 days. 周一 means first stage in that circle, or Monday. 周末 means last stage in that circle, or last day of the week, or Sunday. 末 means last。
To make things clear, 周期 means one cycle。
As I have said above, 周 can means a lot of things. For instance, 彬周 literally means male penis。Thus, you should always use 星期 for a week if you are don't want to embarrass yourself because 星期 means only a week and nothing else.
Dividing a year into 52 weeks was a new concept to Chinese people at the time. Since 星期(指七夕，就是牛郎织女鹊桥相会之日) a common term already existed in China, https://zh.wikipedia.org/wiki/%E6%98%9F%E6%9C%9F it was then used to express a "week".
*The Japanese still use 七曜 for the name of their days in a week - (日曜日, 月曜日, 火曜日 水曜日, 木曜日, 金曜日 and 土曜日)
Meanwhile, there was another common Chinese term 周天 (a circle of) http://baike.baidu.com/item/%E5%91%A8%E5%A4%A9/909115 therefore, 周 was also borrowed to describe "a week"
As for practical usage, 周 and 星期 are interchangeable, but 周 is more formal and 星期 is more colloquial.
Also, the single syllable 周 doesn't take classifier, but 星期 does. We say 一星期 or 一個星期, but we only say 一周, not 一個周
both 周一 and 星期一 mean "Monday", as mentioned before, and they are interchangeable, only 周一 is more formal and 星期一 is more colloquial
there are no differences in meaning ， the only differences is where those words come from.
星期 and 礼拜 are originally from china, but 周 is barely a new word, that caused by western influences
for more information, you can read this article http://www.zybang.com/question/46a3fe5084005470d13f83db003fc4ae.html?fr=iks&word=%D6%DC%BA%CD%D0%C7%C6%DA+%B5%D8%B7%D6%B1%F0&ie=gbk